Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer's Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs

1. Background

Fuzzy set in Ref. [54] by Zadeh (1965), intuitionistic fuzzy sets in Ref. [41] by Atanassov (1986), a first step to a theory of the intuitionistic fuzzy graphs in Ref. [51] by Shannon and Atanassov (1994), a unifying field in logics neutrosophy: neutrosophic probability, set and logic, rehoboth in Ref. [52] by Smarandache (1998), single-valued neutrosophic sets in Ref. [53] by Wang et al. (2010), single-valued neutrosophic graphs in Ref. [45] by Broumi et al. (2016), operations on single-valued neutrosophic graphs in Ref. [37] by Akram and Shahzadi (2017), neutrosophic soft graphs in Ref. [50] by Shah and Hussain (2016), bounds on the average and minimum attendance in preference-based activity scheduling in Ref. [39] by Aronshtam and Ilani (2022), investigating the recoverable robust single machine scheduling problem under interval uncertainty in Ref. [44] by Bold and Goerigk (2022), polyhedra associated with locating-dominating, open locating-dominating and locating total-dominating sets in graphs in Ref. [38] by G. Argiroffo et al. (2022), a Vizing-type result for semi-total domination in Ref. [40] by J. Asplund et al. (2020), total domination cover rubbling in Ref. [42] by R.A. Beeler et al. (2020), on the global total k-domination number of graphs in Ref. [43] by S. Bermudo et al. (2019), maker–breaker total domination game in Ref. [46] by V. Gledel et al. (2020), a new upper bound on the total domination number in graphs with minimum degree six in Ref. [47] by M.A. Henning, and A. Yeo (2021), effect of predomination and vertex removal on the game total domination number of a graph in Ref. [48] by V. Irsic (2019), hardness results of global total k-domination problem in graphs in Ref. [49] by B.S. Panda, and P. Goyal (2021), are studied.

Look at [32,33,34,35,36] for further researches on this topic. See the seminal researches [1,2,3]. The formalization of the notions on the framework of Extreme Failed SuperHyperClique theory, Neutrosophic Failed SuperHyperClique theory, and (Neutrosophic) SuperHyperGraphs theory at [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Two popular research books in Scribd in the terms of high readers, 2638 and 3363 respectively, on neutrosophic science is on [30,31].

2. Neutrosophic Applications in Cancer’s Neutrosophic Recognition toward Neutrosophic Failed SuperHyperStable

For neutrosophic giving the neutrosophic sense about the neutrosophic visions on this neutrosophic event, the neutrosophic Failed SuperHyperStable is neutrosophicly applied in the general neutrosophic forms and the neutrosophic arrangements of the internal neutrosophic venues. Regarding the neutrosophic generality, the next section is introduced.

Definition 1.  ((neutrosophic) Failed SuperHyperStable).

Assume a SuperHyperGraph. Then

$\left(i\right)$

a  Failed SuperHyperStable  $\mathcal{I}\left(NSHG\right)$ for a SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet S of SuperHyperVertices such that there’s a SuperHyperVertex to have a SuperHyperEdge in common;

$\left(ii\right)$

a  neutrosophic Failed SuperHyperStable  ${\mathcal{I}}_n\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common.

Definition 2.  ((neutrosophic)$\delta -$Failed SuperHyperStable).

Assume a SuperHyperGraph. Then

$\left(i\right)$

an  $\delta -$Failed SuperHyperStable  is a maximal of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$

$$\begin{array}{ccc}& & |S\cap N(s\left)\right|>|S\cap (V\setminus N\left(s\right)\left)\right|+\delta ;\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & |S\cap N(s\left)\right|<|S\cap (V\setminus N\left(s\right)\left)\right|+\delta .\hfill \end{array}$$

(2)

The Expression 1, holds if S is an  $\delta -$SuperHyperOffensive  . And the Expression , holds if S is an  $\delta -$SuperHyperDefensive;

$\left(ii\right)$

a   neutrosophic $\delta -$Failed SuperHyperStable  is a maximal neutrosophic of SuperHyperVertices with maximum neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of SuperHyperNeighbors of $s\in S:$

$$\begin{array}{ccc}& & {|S\cap N\left(s\right)|}_{neutrosophic}>{|S\cap (V\setminus N\left(s\right))|}_{neutrosophic}+\delta ;\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & {|S\cap N\left(s\right)|}_{neutrosophic}<{|S\cap (V\setminus N\left(s\right))|}_{neutrosophic}+\delta .\hfill \end{array}$$

(4)

The Expression 3, holds if S is a neutrosophic $\delta -$SuperHyperOffensive . And the Expression , holds if S is a neutrosophic $\delta -$SuperHyperDefensive .

3. General Neutrosophic Results for Cancer’s Neutrosophic Recognition toward Neutrosophic Failed SuperHyperStable

For the Neutrosophic Failed SuperHyperStable, and the Neutrosophic Failed SuperHyperStable, some general results are introduced.

Remark 1. 

Let remind that the Neutrosophic Failed SuperHyperStable is “redefined” on the positions of the alphabets.

Corollary 1. 

Assume Neutrosophic Failed SuperHyperStable. Then

$$\begin{array}{ccc}& & NeutrosophicNeutrosophicFailedSuperHyperStable=\hfill \\ & & \left\{theNeutrosophicFailedSuperHyperStableoftheSuperHyperVertices\right|\hfill \\ & & max|SuperHyperDefensiveSuperHyper\hfill \\ & & {Stable|}_{neutrosophiccardinalityamidthoseNeutrosophicFailedSuperHyperStable.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Corollary 2. 

Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then the notion of Neutrosophic Failed SuperHyperStable and Neutrosophic Failed SuperHyperStable coincide.

Corollary 3. 

Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a Neutrosophic Failed SuperHyperStable if and only if it’s a Failed SuperHyperStable.

Corollary 4. 

Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a strongest SuperHyperCycle if and only if it’s a longest SuperHyperCycle.

Corollary 5. 

Assume SuperHyperClasses of a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then its Neutrosophic Failed SuperHyperStable is its Neutrosophic Failed SuperHyperStable and reversely.

Corollary 6. 

Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel) on the same identical letter of the alphabet. Then its Neutrosophic Failed SuperHyperStable is its Neutrosophic Failed SuperHyperStable and reversely.

Corollary 7. 

Assume a neutrosophic SuperHyperGraph. Then its Neutrosophic Failed SuperHyperStable isn’t well-defined if and only if its Neutrosophic Failed SuperHyperStable isn’t well-defined.

Corollary 8. 

Assume SuperHyperClasses of a neutrosophic SuperHyperGraph. Then its Neutrosophic Failed SuperHyperStable isn’t well-defined if and only if its Neutrosophic Failed SuperHyperStable isn’t well-defined.

Corollary 9. 

Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Neutrosophic Failed SuperHyperStable isn’t well-defined if and only if its Neutrosophic Failed SuperHyperStable isn’t well-defined.

Corollary 10. 

Assume a neutrosophic SuperHyperGraph. Then its Neutrosophic Failed SuperHyperStable is well-defined if and only if its Neutrosophic Failed SuperHyperStable is well-defined.

Corollary 11. 

Assume SuperHyperClasses of a neutrosophic SuperHyperGraph. Then its Neutrosophic Failed SuperHyperStable is well-defined if and only if its Neutrosophic Failed SuperHyperStable is well-defined.

Corollary 12. 

Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Neutrosophic Failed SuperHyperStable is well-defined if and only if its Neutrosophic Failed SuperHyperStable is well-defined.

Proposition 1. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph. Then V is

$\left(i\right):$

the dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

the strong dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

the connected dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

the δ-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

the strong δ-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

the connected δ-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 2. 

Let $NTG:(V,E,\sigma ,\mu )$ be a neutrosophic SuperHyperGraph. Then ∅ is

$\left(i\right):$

the SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

the strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

the connected defensive SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

the δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

the strong δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

the connected δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 3. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph. Then an independent SuperHyperSet is

$\left(i\right):$

the SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

the strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

the connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

the δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

the strong δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

the connected δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 4. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperCycle/SuperHyperPath. Then V is a maximal

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$\mathcal{O}\left(NSHG\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $\mathcal{O}\left(NSHG\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $\mathcal{O}\left(NSHG\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.

Proposition 5. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is a SuperHyperUniform SuperHyperWheel. Then V is a maximal

$\left(i\right):$

dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$\mathcal{O}\left(NSHG\right)$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $\mathcal{O}\left(NSHG\right)$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $\mathcal{O}\left(NSHG\right)$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.

Proposition 6. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperCycle/SuperHyperPath. Then the number of

$\left(i\right):$

the Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

the Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

the connected Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

the $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

the strong $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

the connected $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable.

is one and it’s only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.

Proposition 7. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of

$\left(i\right):$

the dual Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

the dual Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

the dual connected Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

the dual $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

the strong dual $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

the connected dual $\mathcal{O}\left(NSHG\right)$-Neutrosophic Failed SuperHyperStable.

is one and it’s only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.

Proposition 8. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying r with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices is a

$\left(i\right):$

dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 9. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains the half of multiplying r with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart is a

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected δ-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 10. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of

$\left(i\right):$

dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $\frac{\mathcal{O}\left(NSHG\right)}{2}+1$-dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

is one and it’s only $S,$ a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying r with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices. Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.

Proposition 11. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph. The number of connected component is $|V-S|$ if there’s a SuperHyperSet which is a dual

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong 1-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected 1-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 12. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph. Then the number is at most $\mathcal{O}\left(NSHG\right)$ and the neutrosophic number is at most ${\mathcal{O}}_n\left(NSHG\right).$

Proposition 13. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph. Then the number is at most $\mathcal{O}\left(NSHG\right)$ and the neutrosophic number is at most ${\mathcal{O}}_n\left(NSHG\right).$

Proposition 14. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1$ and the neutrosophic number is $min{\Sigma }_{v\in {\{v_1,v_2,\cdots ,v_t\}}_{t>\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}}\subseteq V}\sigma \left(v\right),$ in the setting of dual

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 15. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is $\varnothing .$ The number is 0 and the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of dual

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

0-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong 0-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected 0-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 16. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperComplete. Then there’s no independent SuperHyperSet.

Proposition 17. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperCycle/SuperHyperPath/SuperHyperWheel. The number is $\mathcal{O}(NSHG:(V,E\left)\right)$ and the neutrosophic number is ${\mathcal{O}}_n(NSHG:(V,E)),$ in the setting of a dual

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$\mathcal{O}(NSHG:(V,E\left)\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $\mathcal{O}(NSHG:(V,E\left)\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $\mathcal{O}(NSHG:(V,E\left)\right)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 18. 

Let $NSHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is $\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1$ and the neutrosophic number is $min{\Sigma }_{v\in {\{v_1,v_2,\cdots ,v_t\}}_{t>\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}}\subseteq V}\sigma \left(v\right),$ in the setting of a dual

$\left(i\right):$

SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right):$

strong SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right):$

connected SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right):$

$(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(v\right):$

strong $(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(vi\right):$

connected $(\frac{\mathcal{O}(NSHG:(V,E\left)\right)}{2}+1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 19. 

Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $NSHGs:(V,E)$ neutrosophic SuperHyperGraphs which are from one-type SuperHyperClass which the result is obtained for the individuals. Then the results also hold for the SuperHyperFamily $\mathcal{NSHF}:(V,E)$ of these specific SuperHyperClasses of the neutrosophic SuperHyperGraphs.

Proposition 20. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. If S is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then $\forall v\in V\setminus S,\exists x\in S$ such that

$\left(i\right)$

$v\in N_s\left(x\right);$

$\left(ii\right)$

$vx\in E.$

Proposition 21. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. If S is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then

$\left(i\right)$

S is SuperHyperDominating set;

$\left(ii\right)$

there’s $S\subseteq S^{\prime }$ such that $|S^{\prime }|$ is SuperHyperChromatic number.

Proposition 22. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. Then

$\left(i\right)$

$\Gamma \le \mathcal{O};$

$\left(ii\right)$

${\Gamma }_s\le {\mathcal{O}}_n.$

Proposition 23. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph which is connected. Then

$\left(i\right)$

$\Gamma \le \mathcal{O}-1;$

$\left(ii\right)$

${\Gamma }_s\le {\mathcal{O}}_n-{\Sigma }_{i=1}^3{\sigma }_i\left(x\right).$

Proposition 24. 

Let $NSHG:(V,E)$ be an odd SuperHyperPath. Then

$\left(i\right)$

the SuperHyperSet $S=\{v_2,v_4,\cdots ,v_{n-1}\}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor +1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots ,v_{n-1}\}$;

$\left(iii\right)$

${\Gamma }_s=min\{{\Sigma }_{s\in S=\{v_2,v_4,\cdots ,v_{n-1}\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right),{\Sigma }_{s\in S=\{v_1,v_3,\cdots ,v_{n-1}\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\};$

$\left(iv\right)$

the SuperHyperSets $S_1=\{v_2,v_4,\cdots ,v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots ,v_{n-1}\}$ are only a dual Neutrosophic Failed SuperHyperStable.

Proposition 25. 

Let $NSHG:(V,E)$ be an even SuperHyperPath. Then

$\left(i\right)$

the set $S=\{v_2,v_4,\cdots .v_n\}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots .v_n\}$ and $\{v_1,v_3,\cdots .v_{n-1}\};$

$\left(iii\right)$

${\Gamma }_s=min\{{\Sigma }_{s\in S=\{v_2,v_4,\cdots ,v_n\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right),{\Sigma }_{s\in S=\{v_1,v_3,\cdots .v_{n-1}\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\};$

$\left(iv\right)$

the SuperHyperSets $S_1=\{v_2,v_4,\cdots .v_n\}$ and $S_2=\{v_1,v_3,\cdots .v_{n-1}\}$ are only dual Neutrosophic Failed SuperHyperStable.

Proposition 26. 

Let $NSHG:(V,E)$ be an even SuperHyperCycle. Then

$\left(i\right)$

the SuperHyperSet $S=\{v_2,v_4,\cdots ,v_n\}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots ,v_n\}$ and $\{v_1,v_3,\cdots ,v_{n-1}\};$

$\left(iii\right)$

${\Gamma }_s=min\{{\Sigma }_{s\in S=\{v_2,v_4,\cdots ,v_n\}}\sigma \left(s\right),{\Sigma }_{s\in S=\{v_1,v_3,\cdots ,v_{n-1}\}}\sigma \left(s\right)\};$

$\left(iv\right)$

the SuperHyperSets $S_1=\{v_2,v_4,\cdots ,v_n\}$ and $S_2=\{v_1,v_3,\cdots ,v_{n-1}\}$ are only dual Neutrosophic Failed SuperHyperStable.

Proposition 27. 

Let $NSHG:(V,E)$ be an odd SuperHyperCycle. Then

$\left(i\right)$

the SuperHyperSet $S=\{v_2,v_4,\cdots ,v_{n-1}\}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor +1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots ,v_{n-1}\}$;

$\left(iii\right)$

${\Gamma }_s=min\{{\Sigma }_{s\in S=\{v_2,v_4,\cdots .v_{n-1}\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right),{\Sigma }_{s\in S=\{v_1,v_3,\cdots .v_{n-1}\}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\};$

$\left(iv\right)$

the SuperHyperSets $S_1=\{v_2,v_4,\cdots .v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots .v_{n-1}\}$ are only dual Neutrosophic Failed SuperHyperStable.

Proposition 28. 

Let $NSHG:(V,E)$ be SuperHyperStar. Then

$\left(i\right)$

the SuperHyperSet $S=\left\{c\right\}$ is a dual maximal Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =1;$

$\left(iii\right)$

${\Gamma }_s={\Sigma }_{i=1}^3{\sigma }_i\left(c\right);$

$\left(iv\right)$

the SuperHyperSets $S=\left\{c\right\}$ and $S\subset S^{\prime }$ are only dual Neutrosophic Failed SuperHyperStable.

Proposition 29. 

Let $NSHG:(V,E)$ be SuperHyperWheel. Then

$\left(i\right)$

the SuperHyperSet $S=\{v_1,v_3\}\cup {\{v_6,v_9\cdots ,v_{i+6},\cdots ,v_n\}}_{i=1}^{6+3(i-1)\le n}$ is a dual maximal SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =|\{v_1,v_3\}\cup {\{v_6,v_9\cdots ,v_{i+6},\cdots ,v_n\}}_{i=1}^{6+3(i-1)\le n}|;$

$\left(iii\right)$

${\Gamma }_s={\Sigma }_{\{v_1,v_3\}\cup {\{v_6,v_9\cdots ,v_{i+6},\cdots ,v_n\}}_{i=1}^{6+3(i-1)\le n}}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right);$

$\left(iv\right)$

the SuperHyperSet $\{v_1,v_3\}\cup {\{v_6,v_9\cdots ,v_{i+6},\cdots ,v_n\}}_{i=1}^{6+3(i-1)\le n}$ is only a dual maximal SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 30. 

Let $NSHG:(V,E)$ be an odd SuperHyperComplete. Then

$\left(i\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor +1;$

$\left(iii\right)$

${\Gamma }_s=min{\left\{{\Sigma }_{s\in S}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\right\}}_{S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}};$

$\left(iv\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}$ is only a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 31. 

Let $NSHG:(V,E)$ be an even SuperHyperComplete. Then

$\left(i\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor ;$

$\left(iii\right)$

${\Gamma }_s=min{\left\{{\Sigma }_{s\in S}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\right\}}_{S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }};$

$\left(iv\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }$ is only a dual maximal SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 32. 

Let $\mathcal{NSHF}:(V,E)$ be a m-SuperHyperFamily of neutrosophic SuperHyperStars with common neutrosophic SuperHyperVertex SuperHyperSet. Then

$\left(i\right)$

the SuperHyperSet $S=\{c_1,c_2,\cdots ,c_m\}$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF};$

$\left(ii\right)$

$\Gamma =m$ for $\mathcal{NSHF}:(V,E);$

$\left(iii\right)$

${\Gamma }_s={\Sigma }_{i=1}^m{\Sigma }_{j=1}^3{\sigma }_j\left(c_i\right)$ for $\mathcal{NSHF}:(V,E);$

$\left(iv\right)$

the SuperHyperSets $S=\{c_1,c_2,\cdots ,c_m\}$ and $S\subset S^{\prime }$ are only dual Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF}:(V,E).$

Proposition 33. 

Let $\mathcal{NSHF}:(V,E)$ be an m-SuperHyperFamily of odd SuperHyperComplete SuperHyperGraphs with common neutrosophic SuperHyperVertex SuperHyperSet. Then

$\left(i\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}$ is a dual maximal SuperHyperDefensive Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF};$

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor +1$ for $\mathcal{NSHF}:(V,E);$

$\left(iii\right)$

${\Gamma }_s=min{\left\{{\Sigma }_{s\in S}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\right\}}_{S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}}$ for $\mathcal{NSHF}:(V,E);$

$\left(iv\right)$

the SuperHyperSets $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor +1}$ are only a dual maximal Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF}:(V,E).$

Proposition 34. 

Let $\mathcal{NSHF}:(V,E)$ be a m-SuperHyperFamily of even SuperHyperComplete SuperHyperGraphs with common neutrosophic SuperHyperVertex SuperHyperSet. Then

$\left(i\right)$

the SuperHyperSet $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }$ is a dual SuperHyperDefensive Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF}:(V,E);$

$\left(ii\right)$

$\Gamma =\lfloor \frac{n}{2}\rfloor$ for $\mathcal{NSHF}:(V,E);$

$\left(iii\right)$

${\Gamma }_s=min{\left\{{\Sigma }_{s\in S}{\Sigma }_{i=1}^3{\sigma }_i\left(s\right)\right\}}_{S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }}$ for $\mathcal{NSHF}:(V,E);$

$\left(iv\right)$

the SuperHyperSets $S={\left\{v_i\right\}}_{i=1}^{\lfloor \frac{n}{2}\rfloor }$ are only dual maximal Neutrosophic Failed SuperHyperStable for $\mathcal{NSHF}:(V,E).$

Proposition 35. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. Then following statements hold;

$\left(i\right)$

if $s\ge t$ and a SuperHyperSet S of SuperHyperVertices is an t-SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then S is an s-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

if $s\le t$ and a SuperHyperSet S of SuperHyperVertices is a dual t-SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then S is a dual s-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 36. 

Let $NSHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. Then following statements hold;

$\left(i\right)$

if $s\ge t+2$ and a SuperHyperSet S of SuperHyperVertices is an t-SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then S is an s-SuperHyperPowerful Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

if $s\le t$ and a SuperHyperSet S of SuperHyperVertices is a dual t-SuperHyperDefensive Neutrosophic Failed SuperHyperStable, then S is a dual s-SuperHyperPowerful Neutrosophic Failed SuperHyperStable.

Proposition 37. 

Let $NSHG:(V,E)$ be a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then following statements hold;

$\left(i\right)$

if $\forall a\in S,|N_s\left(a\right)\cap S|<\lfloor \frac{r}{2}\rfloor +1,$ then $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>\lfloor \frac{r}{2}\rfloor +1,$ then $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

if $\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is an r-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is a dual r-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 38. 

Let $NSHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then following statements hold;

$\left(i\right)$

$\forall a\in S,|N_s\left(a\right)\cap S|<\lfloor \frac{r}{2}\rfloor +1$ if $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>\lfloor \frac{r}{2}\rfloor +1$ if $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

$\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is an r-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is a dual r-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 39. 

Let $NSHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then following statements hold;

$\left(i\right)$

$\forall a\in S,|N_s\left(a\right)\cap S|<\lfloor \frac{\mathcal{O}-1}{2}\rfloor +1$ if $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>\lfloor \frac{\mathcal{O}-1}{2}\rfloor +1$ if $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

$\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 40. 

Let $NSHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then following statements hold;

$\left(i\right)$

if $\forall a\in S,|N_s\left(a\right)\cap S|<\lfloor \frac{\mathcal{O}-1}{2}\rfloor +1,$ then $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>\lfloor \frac{\mathcal{O}-1}{2}\rfloor +1,$ then $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

if $\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 41. 

Let $NSHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is SuperHyperCycle. Then following statements hold;

$\left(i\right)$

$\forall a\in S,|N_s\left(a\right)\cap S|<2$ if $NSHG:(V,E))$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>2$ if $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

$\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

$\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0$ if $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

Proposition 42. 

Let $NSHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is SuperHyperCycle. Then following statements hold;

$\left(i\right)$

if $\forall a\in S,|N_s\left(a\right)\cap S|<2,$ then $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(ii\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap S|>2,$ then $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iii\right)$

if $\forall a\in S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is an 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable;

$\left(iv\right)$

if $\forall a\in V\setminus S,|N_s\left(a\right)\cap V\setminus S|=0,$ then $NSHG:(V,E)$ is a dual 2-SuperHyperDefensive Neutrosophic Failed SuperHyperStable.

4. Neutrosophic Motivation and Neutrosophic Contributions

In this research, there are some Neutrosophic ideas in the featured Neutrosophic frameworks of Neutrosophic motivations. I try to bring the Neutrosophic motivations in the narrative Neutrosophic ways.

Question 1. 

How to define the Neutrosophic SuperHyperNotions and to do research on them to find the “ Neutrosophic amount of Neutrosophic” of either individual of Neutrosophic cells or the Neutrosophic groups of Neutrosophic cells based on the fixed Neutrosophic cell or the fixed Neutrosophic group of Neutrosophic cells, extensively, the “Neutrosophic amount of Neutrosophic” based on the fixed Neutrosophic groups of Neutrosophic cells or the fixed Neutrosophic groups of Neutrosophic group of Neutrosophic cells?

Question 2. 

What are the best Neutrosophic descriptions for the “Cancer’s Neutrosophic Recognitions” in Neutrosophic terms of these messy and dense Neutrosophic SuperHyperModels where Neutrosophic embedded notions are Neutrosophicly illustrated?

It’s Neutrosophic motivation to find Neutrosophic notions to use in this dense Neutrosophic model is titled “Neutrosophic SuperHyperGraphs”. Thus it motivates us to define different types of “Neutrosophic” and “Neutrosophic” on “SuperHyperGraph” and “Neutrosophic SuperHyperGraph”.

5. Neutrosophic Failed SuperHyperStable in Some Neutrosophic Situations for Cancer without any names or any Neutrosophic specific classes

Example 1. 

Assume the neutrosophic SuperHyperGraph s in the Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20.

• On the Figure 1, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. $E_1$ and $E_3$ neutrosophic Failed SuperHyperStable are some empty neutrosophic SuperHyperEdges but $E_2$ is a loop neutrosophic SuperHyperEdge and $E_4$ is a neutrosophic SuperHyperEdge. Thus in the terms of neutrosophic SuperHyperNeighbor, there’s only one neutrosophic SuperHyperEdge, namely, $E_4.$ The neutrosophic SuperHyperVertex, $V_3$ is isolated means that there’s no neutrosophic SuperHyperEdge has it as an endpoint. Thus neutrosophic SuperHyperVertex, $V_3,$ is contained in every given neutrosophic Failed SuperHyperStable. All the following SuperHyperSet of neutrosophic SuperHyperVertices is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. $\{V_3,V_1,V_2\}.$ The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is corresponded to a neutrosophic Failed SuperHyperStable $\mathcal{I}\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only three neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex. But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is corresponded to a neutrosophic Failed SuperHyperStable $\mathcal{I}\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and they are corresponded to a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_3,V_1,V_2\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_3,V_1,V_2\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_3,V_1,V_2\},$ is the SuperHyperSet, $\{V_3,V_1,V_2\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ It’s interesting to mention that the only obvious simple type-SuperHyperSet of the neutrosophic neutrosophic Failed SuperHyperStable amid those obvious simple type-SuperHyperSets of the neutrosophic Failed SuperHyperStable, is only $\{V_3,V_4,V_2\}.$
• On the Figure 2, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. $E_1$ and $E_3$ neutrosophic Failed SuperHyperStable are some empty neutrosophic SuperHyperEdges but $E_2$ is a loop neutrosophic SuperHyperEdge and $E_4$ is a neutrosophic SuperHyperEdge. Thus in the terms of neutrosophic SuperHyperNeighbor, there’s only one neutrosophic SuperHyperEdge, namely, $E_4.$ The neutrosophic SuperHyperVertex, $V_3$ is isolated means that there’s no neutrosophic SuperHyperEdge has it as an endpoint. Thus neutrosophic SuperHyperVertex, $V_3,$ is contained in every given neutrosophic Failed SuperHyperStable. All the following SuperHyperSet of neutrosophic SuperHyperVertices is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. $\{V_3,V_1,V_2\}.$ The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is corresponded to a neutrosophic Failed SuperHyperStable $\mathcal{I}\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only three neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex. But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1,V_2\},$ is corresponded to a neutrosophic Failed SuperHyperStable $\mathcal{I}\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and they are corresponded to a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_3,V_1,V_2\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_3,V_1,V_2\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_3,V_1,V_2\},$ is the SuperHyperSet, $\{V_3,V_1,V_2\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ It’s interesting to mention that the only obvious simple type-SuperHyperSet of the neutrosophic neutrosophic Failed SuperHyperStable amid those obvious simple type-SuperHyperSets of the neutrosophic Failed SuperHyperStable, is only $\{V_3,V_4,V_1\}.$
• On the Figure 3, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. $E_1,E_2$ and $E_3$ are some empty neutrosophic SuperHyperEdges but $E_4$ is a neutrosophic SuperHyperEdge. Thus in the terms of neutrosophic SuperHyperNeighbor, there’s only one neutrosophic SuperHyperEdge, namely, $E_4.$ The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_2\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_2\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only two neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_2\},$ doesn’t have less than two neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_2\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_2\},$ is corresponded to a neutrosophic Failed SuperHyperStable $\mathcal{I}\left(NSHG\right)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and they are neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSets, $\{V_3,V_2\},$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_3,V_2\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_3,V_2\},$ is the SuperHyperSet, $\{V_3,V_2\},$ don’t include only more than one neutrosophic SuperHyperVertex in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ It’s interesting to mention that the only obvious simple type-SuperHyperSets of the neutrosophic neutrosophic Failed SuperHyperStable amid those obvious simple type-SuperHyperSets of the neutrosophic Failed SuperHyperStable, is only $\{V_3,V_2\}.$
• On the Figure 4, the neutrosophic SuperHyperNotion, namely, a neutrosophic Failed SuperHyperStable, is up. There’s no empty neutrosophic SuperHyperEdge but $E_3$ are a loop neutrosophic SuperHyperEdge on $\left\{F\right\},$ and there are some neutrosophic SuperHyperEdges , namely, $E_1$ on $\{H,V_1,V_3\},$ alongside $E_2$ on $\{O,H,V_4,V_3\}$ and $E_4,E_5$ on $\{N,V_1,V_2,V_3,F\}.$ The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_4,V_1\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_4,V_1\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only three neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex since it doesn’t form any kind of pairs titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_4,V_1\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_4,V_1\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_4,V_1\},$ is the SuperHyperSet Ss of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_4,V_1\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_4,V_1\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_2,V_4,V_1\},$ is a SuperHyperSet, $\{V_2,V_4,V_1\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 5, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only one neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex thus it doesn’t form any kind of pairs titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_6,V_9,V_{15},V_{10}\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. and it’s neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_6,V_9,V_{15},V_{10}\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ is a SuperHyperSet, $\{V_2,V_6,V_9,V_{15},V_{10}\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ is mentioned as the neutrosophic SuperHyperModel $NSHG:(V,E)$ in the Figure 5.
• On the Figure 6, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only one neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex doesn’t form any kind of pairs titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  Thus the non-obvious neutrosophic Failed SuperHyperStable,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is a SuperHyperSet,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ with a illustrated neutrosophic SuperHyperModel ing of the Figure 6.
• On the Figure 7, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’s only one neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_5,V_9,V_7\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_9,V_7\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,$\{V_2,V_5,V_9,V_7\},$ is a SuperHyperSet, $\{V_2,V_5,V_9,V_7\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ of depicted neutrosophic SuperHyperModel as the Figure 7.
• On the Figure 8, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’s only one neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_9,V_7\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_5,V_9,V_7\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_9,V_7\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,$\{V_2,V_5,V_9,V_7\},$ is a SuperHyperSet, $\{V_2,V_5,V_9,V_7\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ of dense neutrosophic SuperHyperModel as the Figure 8.
• On the Figure 9, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only only neutrosophic SuperHyperVertex inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only one neutrosophic SuperHyperVertex doesn’t form any kind of pairs titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\}.\hfill \end{array}$$
  Thus the non-obvious neutrosophic Failed SuperHyperStable,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  is a SuperHyperSet,

  $$\begin{array}{ccc}& & \{V_2,V_4,V_6,V_8,V_{10},\hfill \\ & & V_{22},V_{19},V_{17},V_{15},V_{13},V_{11}\},\hfill \end{array}$$
  doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic neutrosophic SuperHyperGraph $NSHG:(V,E)$ with a messy neutrosophic SuperHyperModel ing of the Figure 9.
• On the Figure 10, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_8,V_7\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_8,V_7\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_8\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_8,V_7\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_8,V_7\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_5,V_8,V_7\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_8,V_7\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_8,V_7\},$ is a SuperHyperSet, $\{V_2,V_5,V_8,V_7\},$ doesn’t include only more than one neutrosophic SuperHyperVertex in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ of highly-embedding-connected neutrosophic SuperHyperModel as the Figure 10.
• On the Figure 11, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only less than one neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices don’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_5,V_6\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_6\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_6\},$ is a SuperHyperSet, $\{V_2,V_5,V_6\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 12, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ is the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and they are neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_4,V_5,V_6,V_9,V_{10},V_2\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,$\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ is a SuperHyperSet, $\{V_4,V_5,V_6,V_9,V_{10},V_2\},$ doesn’t include only more than one neutrosophic SuperHyperVertex in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ in highly-multiple-connected-style neutrosophic SuperHyperModel On the Figure 12.
• On the Figure 13, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re not only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices don’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_2,V_5,V_6\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_2,V_5,V_6\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_6\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_2,V_5,V_6\},$ is a SuperHyperSet, $\{V_2,V_5,V_6\},$ does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 14, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_3,V_1\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_3,V_1\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_3,V_1\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_3,V_1\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_3,V_1\},$ is a SuperHyperSet, $\{V_3,V_1\},$ does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 15, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_5,V_2,V_6,V_4\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_5,V_2,V_6,V_4\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices ,$\{V_5,V_2,V_6,V_4\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_5,V_2,V_6,V_4\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_5,V_2,V_6,V_4\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_5,V_2,V_6,V_4\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_5,V_2,V_6,V_4\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_5,V_2,V_6,V_4\},$ is a SuperHyperSet, $\{V_5,V_2,V_6,V_4\},$ doesn’t include only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ as Linearly-Connected neutrosophic SuperHyperModel On the Figure 15.
• On the Figure 16, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is a SuperHyperSet, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 17, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet,$\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is a SuperHyperSet, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$ as Linearly-over-packed neutrosophic SuperHyperModel is featured On the Figure 17.
• On the Figure 18, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\}.$ Thus the non-obvious neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ is a SuperHyperSet, $\{V_1,V_3,V_7,V_{13},V_{22},V_{18}\},$ does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E)$
• On the Figure 19, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices ,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}}.$$
  Thus the non-obvious neutrosophic Failed SuperHyperStable,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is a SuperHyperSet,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$
• On the Figure 20, the neutrosophic SuperHyperNotion, namely, neutrosophic Failed SuperHyperStable, is up. There’s neither empty neutrosophic SuperHyperEdge nor loop neutrosophic SuperHyperEdge. The SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. The SuperHyperSet of the neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There’re only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious neutrosophic Failed SuperHyperStable is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is a SuperHyperSet includes only less than two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled to neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ But the SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  doesn’t have less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet. Thus the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable is up. To sum them up, the SuperHyperSet of neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the non-obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable. Since the SuperHyperSet of the neutrosophic SuperHyperVertices,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is the SuperHyperSet Ss of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common and it’s a neutrosophic Failed SuperHyperStable. Since it’s the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. There aren’t only less than two neutrosophic SuperHyperVertices inside the intended SuperHyperSet,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}}.$$
  Thus the non-obvious neutrosophic Failed SuperHyperStable,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable,

  $${\left\{\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\right\}}_{\mathit{the}\mathit{number}\mathit{of}\mathit{neutrosophic}\mathit{SuperHyperEdges}},$$
  is a SuperHyperSet, does includes only less than two neutrosophic SuperHyperVertices in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$

Figure 1. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 1. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Proposition 43. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Then in the worst case, literally, $V\setminus V\setminus \{x,z\},$ is a neutrosophic Failed SuperHyperStable. In other words, the least neutrosophic cardinality, the lower sharp bound for the neutrosophic cardinality, of a neutrosophic Failed SuperHyperStable is the neutrosophic cardinality of $V\setminus V\setminus \{x,z\}.$

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. □

Figure 2. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 2. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 3. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 3. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 4. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 4. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 5. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 5. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 6. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 6. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 7. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 7. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 8. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 8. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 9. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 9. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 10. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 10. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 11. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 11. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 12. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 12. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 13. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 13. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 14. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 14. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 15. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 15. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 16. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 16. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 17. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 17. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 18. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 18. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 19. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 19. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 20. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Figure 20. The neutrosophic SuperHyperGraph s Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 1

Proposition 44. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Then the neutrosophic number of neutrosophic Failed SuperHyperStable has, the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality, is the neutrosophic neutrosophic cardinality of $V\setminus V\setminus \{x,z\}$ if there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality .

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ the neutrosophic number of neutrosophic Failed SuperHyperStable has, the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality, is the neutrosophic neutrosophic cardinality of $V\setminus V\setminus \{x,z\}$ if there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality . □

Proposition 45. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ If a neutrosophic SuperHyperEdge has z neutrosophic SuperHyperVertices, then $z-2$ number of those interior neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge exclude to any neutrosophic Failed SuperHyperStable.

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Let a neutrosophic SuperHyperEdge has z neutrosophic SuperHyperVertices. Consider $z-2$ number of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperEdge has z neutrosophic SuperHyperVertices, then $z-2$ number of those interior neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge exclude to any neutrosophic Failed SuperHyperStable. □

Proposition 46. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ There’s only one neutrosophic SuperHyperEdge has only less than three distinct interior neutrosophic SuperHyperVertices inside of any given neutrosophic Failed SuperHyperStable. In other words, there’s only an unique neutrosophic SuperHyperEdge has only two distinct neutrosophic SuperHyperVertices in a neutrosophic Failed SuperHyperStable .

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider some numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices . Consider there’s neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ there’s only one neutrosophic SuperHyperEdge has only less than three distinct interior neutrosophic SuperHyperVertices inside of any given neutrosophic Failed SuperHyperStable. In other words, there’s only an unique neutrosophic SuperHyperEdge has only two distinct neutrosophic SuperHyperVertices in a neutrosophic Failed SuperHyperStable . □

Proposition 47. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The all interior neutrosophic SuperHyperVertices belong to any neutrosophic Failed SuperHyperStable if for any of them, there’s no other corresponded neutrosophic SuperHyperVertex such that the two interior neutrosophic SuperHyperVertices are mutually neutrosophic SuperHyperNeighbors with an exception once.

Proof. 

Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ the all interior neutrosophic SuperHyperVertices belong to any neutrosophic Failed SuperHyperStable if for any of them, there’s no other corresponded neutrosophic SuperHyperVertex such that the two interior neutrosophic SuperHyperVertices are mutually neutrosophic SuperHyperNeighbors with an exception once. □

Proposition 48. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The any neutrosophic Failed SuperHyperStable only contains all interior neutrosophic SuperHyperVertices and all exterior neutrosophic SuperHyperVertices where there’s any of them has no neutrosophic SuperHyperNeighbors in and there’s no neutrosophic SuperHyperNeighborhoods in with an exception once but everything is possible about neutrosophic SuperHyperNeighborhoods and neutrosophic SuperHyperNeighbors out.

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ the any neutrosophic Failed SuperHyperStable only contains all interior neutrosophic SuperHyperVertices and all exterior neutrosophic SuperHyperVertices where there’s any of them has no neutrosophic SuperHyperNeighbors in and there’s no neutrosophic SuperHyperNeighborhoods in with an exception once but everything is possible about neutrosophic SuperHyperNeighborhoods and neutrosophic SuperHyperNeighbors out. □

Remark 2. 

The words “ neutrosophic Failed SuperHyperStable” and “SuperHyperDominating” both refer to the maximum type-style. In other words, they both refer to the maximum number and the SuperHyperSet with the maximum neutrosophic cardinality.

Proposition 49. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Consider a SuperHyperDominating. Then a neutrosophic Failed SuperHyperStable is either out with one additional member.

Proof. 

Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Consider a SuperHyperDominating. By applying the Proposition 48, the results are up. Thus on a connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ and in a SuperHyperDominating, a neutrosophic Failed SuperHyperStable is either out with one additional member. □

6. Neutrosophic Results on in Some Specific Neutrosophic Situations Neutrosophicly Titled Neutrosophic SuperHyperClasses

Proposition 50. 

Assume a connected neutrosophic SuperHyperPath $NSHP:(V,E).$ Then a neutrosophic Failed SuperHyperStable-style with the maximum neutrosophic SuperHyperCardinality is a SuperHyperSet of the interior neutrosophic SuperHyperVertices .

Proposition 51. 

Assume a connected neutrosophic SuperHyperPath $NSHP:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices from the common neutrosophic SuperHyperEdges excluding only two interior neutrosophic SuperHyperVertices from the common neutrosophic SuperHyperEdges . A neutrosophic Failed SuperHyperStable has the number of all the interior neutrosophic SuperHyperVertices minus their neutrosophic SuperHyperNeighborhoods plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathit{The}\mathit{number}-\mathit{of}-\mathit{all}\hfill \\ & & -\mathit{the}-\mathit{SuperHyperVertices}\hfill \\ & & -\mathit{minus}-\mathit{SuperHyperNeighborhoods}-\mathit{plus}-\mathit{one}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}|min|\mathit{the}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\mathit{SuperHyperVertices}\mathit{with}\mathit{only}\hfill \\ & & \mathit{two}\mathit{exceptions}\hfill \\ & & \mathit{in}\mathit{the}\mathit{form}\mathit{of}\hfill \\ & & \mathit{interior}\hfill \\ & & \mathit{SuperHyperVertices}\hfill \\ & & \mathit{excluding}\mathit{one}\mathit{from}\mathit{common}\hfill \\ & & {\mathit{SuperHyperEdge}.|}_{\mathit{neutrosophic}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Proof. 

Assume a connected neutrosophic SuperHyperPath $NSHP:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperPath $NSHP:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices from the common neutrosophic SuperHyperEdges excluding only two interior neutrosophic SuperHyperVertices from the common neutrosophic SuperHyperEdges . A neutrosophic Failed SuperHyperStable has the number of all the interior neutrosophic SuperHyperVertices minus their neutrosophic SuperHyperNeighborhoods plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathit{The}\mathit{number}-\mathit{of}-\mathit{all}\hfill \\ & & -\mathit{the}-\mathit{SuperHyperVertices}\hfill \\ & & -\mathit{minus}-\mathit{SuperHyperNeighborhoods}-\mathit{plus}-\mathit{one}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}|min|\mathit{the}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\mathit{SuperHyperVertices}\mathit{with}\mathit{only}\hfill \\ & & \mathit{two}\mathit{exceptions}\hfill \\ & & \mathit{in}\mathit{the}\mathit{form}\mathit{of}\hfill \\ & & \mathit{interior}\hfill \\ & & \mathit{SuperHyperVertices}\hfill \\ & & \mathit{excluding}\mathit{one}\mathit{from}\mathit{common}\hfill \\ & & {\mathit{SuperHyperEdge}.|}_{\mathit{neutrosophic}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 2. 

In the Figure 21, the connected neutrosophic SuperHyperPath $NSHP:(V,E),$ is highlighted and featured.

By using the Figure 21 and the Table 1, the neutrosophic SuperHyperPath is obtained.

The SuperHyperSet, $\{V_{27},V_2,V_7,V_{12},V_{22},V_{25}\},$ of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperPath $NSHP:(V,E),$ in the neutrosophic SuperHyperModel 21, is the neutrosophic Failed SuperHyperStable.

Proposition 52. 

Assume a connected neutrosophic SuperHyperCycle $NSHC:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices from the same neutrosophic SuperHyperNeighborhoods excluding one neutrosophic SuperHyperVertex. A neutrosophic Failed SuperHyperStable has the number of all the neutrosophic SuperHyperEdges plus one and the lower bound is the half number of all the neutrosophic SuperHyperEdges plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathit{The}\mathit{number}-\mathit{of}-\mathit{all}\hfill \\ & & -\mathit{the}-\mathit{SuperHyperEdges}\hfill \\ & & -\mathit{plus}-\mathit{one}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}|min|\mathit{the}\mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}\mathit{with}\mathit{only}\hfill \\ & & \mathit{all}\mathit{exceptions}\mathit{in}\mathit{the}\mathit{form}\mathit{of}\mathit{interior}\mathit{SuperHyperVertices}\hfill \\ & & \mathit{excluding}\mathit{one}\hfill \\ & & \mathit{neutrosophic}\mathit{SuperHyperVertex}\hfill \\ & & \mathit{from}\mathit{same}\hfill \\ & & \mathit{neutrosophic}\hfill \\ & & {\mathit{SuperHyperEdge}.|}_{\mathit{neutrosophic}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Figure 21. A neutrosophic SuperHyperPath Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 2

Figure 21. A neutrosophic SuperHyperPath Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 2

Proof. 

Assume a connected neutrosophic SuperHyperCycle $NSHC:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperCycle $NSHC:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices from the same neutrosophic SuperHyperNeighborhoods excluding one neutrosophic SuperHyperVertex. A neutrosophic Failed SuperHyperStable has the number of all the neutrosophic SuperHyperEdges plus one and the lower bound is the half number of all the neutrosophic SuperHyperEdges plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathrm{The}\mathrm{number}-\mathrm{of}-\mathrm{all}\hfill \\ & & -\mathrm{the}-\mathrm{SuperHyperEdges}\hfill \\ & & -\mathrm{plus}-\mathrm{one}\hfill \\ & & \mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\hfill \\ & & \mathrm{SuperHyperVertices}|min|\mathrm{the}\mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\hfill \\ & & \mathrm{SuperHyperVertices}\mathrm{with}\mathrm{only}\hfill \\ & & \mathrm{all}\mathrm{exceptions}\mathrm{in}\mathrm{the}\mathrm{form}\mathrm{of}\mathrm{interior}\mathrm{SuperHyperVertices}\hfill \\ & & \mathrm{excluding}\mathrm{one}\hfill \\ & & \mathrm{neutrosophic}\mathrm{SuperHyperVertex}\hfill \\ & & \mathrm{from}\mathrm{same}\hfill \\ & & \mathrm{neutrosophic}\hfill \\ & & {\mathrm{SuperHyperEdge}.|}_{\mathrm{neutrosophic}\mathrm{cardinality}\mathrm{amid}\mathrm{those}\mathrm{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 3. 

In the Figure 22, the connected neutrosophic SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured.

By using the Figure 22 and the Table 2, the neutrosophic SuperHyperCycle is obtained.

The obtained SuperHyperSet, by the Algorithm in previous result, of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperCycle $NSHC:(V,E),$ in the neutrosophic SuperHyperModel 22,

$$\begin{array}{ccc}& & \{\{P_{13},J_{13},K_{13},H_{13}\},\hfill \\ & & \{Z_{13},W_{13},V_{13}\},\{U_{14},T_{14},R_{14},S_{14}\},\hfill \\ & & \{P_{15},J_{15},K_{15},R_{15}\},\hfill \\ & & \{J_5,O_5,K_5,L_5\},\{J_5,O_5,K_5,L_5\},V_3,\hfill \\ & & \{U_6,H_7,J_7,K_7,O_7,L_7,P_7\},\{T_8,U_8,V_8,S_8\},\hfill \\ & & \{T_9,K_9,J_9\},\{H_{10},J_{10},E_{10},R_{10},W_9\},\hfill \\ & & \{S_{11},R_{11},O_{11},L_{11}\},\hfill \\ & & \{U_{12},V_{12},W_{12},Z_{12},O_{12}\},\hfill \\ & & \{S_7,T_7,R_7,U_7\}\},\hfill \end{array}$$

is the neutrosophic Failed SuperHyperStable.

Proposition 53. 

Assume a connected neutrosophic SuperHyperStar $NSHS:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices, excluding the SuperHyperCenter, with only all exceptions in the form of interior neutrosophic SuperHyperVertices from common neutrosophic SuperHyperEdge, excluding only one neutrosophic SuperHyperVertex. A neutrosophic Failed SuperHyperStable has the number of the neutrosophic cardinality of the second SuperHyperPart plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathit{The}\mathit{number}-\mathit{of}-\mathit{all}\hfill \\ & & -\mathit{the}-\mathit{neutrosophic}-\hfill \\ & & \mathit{cardinality}-\mathit{of}-\mathit{sec}\mathit{ond}-\mathit{SuperHyperPart}-\mathit{plus}-\mathit{one}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}|min|\mathit{the}\mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{SuperHyperVertices}\mathit{with}\mathit{only}\hfill \\ & & \mathit{two}\mathit{exceptions}\mathit{in}\hfill \\ & & \mathit{the}\mathit{form}\mathit{of}\hfill \\ & & \mathit{interior}\hfill \\ & & \mathit{SuperHyperVertices},\hfill \\ & & \mathit{excluding}\mathit{one}\hfill \\ & & \mathit{SuperHyperVertex}\hfill \\ & & \mathit{and}\mathit{the}\mathit{SuperHyperCenter},\hfill \\ & & \mathit{from}\mathit{any}\hfill \\ & & {\mathit{given}\mathit{SuperHyperEdge}.|}_{\mathit{neutrosophic}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Proof. 

Assume a connected neutrosophic SuperHyperStar $NSHS:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperStar $NSHS:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices, excluding the SuperHyperCenter, with only all exceptions in the form of interior neutrosophic SuperHyperVertices from common neutrosophic SuperHyperEdge, excluding only one neutrosophic SuperHyperVertex. A neutrosophic Failed SuperHyperStable has the number of the neutrosophic cardinality of the second SuperHyperPart plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathrm{The}\mathrm{number}-\mathrm{of}-\mathrm{all}\hfill \\ & & -\mathrm{the}-\mathrm{neutrosophic}-\hfill \\ & & \mathrm{cardinality}-\mathrm{of}-\sec \mathrm{ond}-\mathrm{SuperHyperPart}-\mathrm{plus}-\mathrm{one}\hfill \\ & & \mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\hfill \\ & & \mathrm{SuperHyperVertices}|min|\mathrm{the}\mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\hfill \\ & & \mathrm{SuperHyperVertices}\mathrm{with}\mathrm{only}\hfill \\ & & \mathrm{two}\mathrm{exceptions}\mathrm{in}\hfill \\ & & \mathrm{the}\mathrm{form}\mathrm{of}\hfill \\ & & \mathrm{interior}\hfill \\ & & \mathrm{SuperHyperVertices},\hfill \\ & & \mathrm{excluding}\mathrm{one}\hfill \\ & & \mathrm{SuperHyperVertex}\hfill \\ & & \mathrm{and}\mathrm{the}\mathrm{SuperHyperCenter},\hfill \\ & & \mathrm{from}\mathrm{any}\hfill \\ & & {\mathrm{given}\mathrm{SuperHyperEdge}.|}_{\mathrm{neutrosophic}\mathrm{cardinality}\mathrm{amid}\mathrm{those}\mathrm{SuperHyperSets}.}\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 4. 

In the Figure 23, the connected neutrosophic SuperHyperStar $NSHS:(V,E),$ is highlighted and featured.

By using the Figure 23 and the Table 3, the neutrosophic SuperHyperStar is obtained.

The obtained SuperHyperSet, by the Algorithm in previous result, of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperStar $NSHS:(V,E),$ in the neutrosophic SuperHyperModel 23,

$$\begin{array}{ccc}& & \{\{V_{14},O_{14},U_{14}\},\hfill \\ & & \{W_{14},D_{15},Z_{14},C_{15},E_{15}\},\hfill \\ & & \{P_3,O_3,R_3,L_3,S_3\},\{P_2,T_2,S_2,R_2,O_2\},\hfill \\ & & \{O_6,O_7,K_7,P_6,H_7,J_7,E_7,L_7\},\hfill \\ & & \{J_8,Z_{10},W_{10},V_{10}\},\{W_{11},V_{11},Z_{11},C_{12}\},\hfill \\ & & \{U_{13},T_{13},R_{13},S_{13}\},\left\{H_{13}\right\},\hfill \\ & & \{E_{13},D_{13},C_{13},Z_{12}\},\}\hfill \end{array}$$

is the neutrosophic Failed SuperHyperStable.

Proposition 54. 

Assume a connected neutrosophic SuperHyperBipartite $NSHB:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices titled neutrosophic SuperHyperNeighbors with only one exception. A neutrosophic Failed SuperHyperStable has the number of the neutrosophic cardinality of the first SuperHyperPart multiplies with the neutrosophic cardinality of the second SuperHyperPart plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathit{The}\mathit{number}-\mathit{of}-\mathit{all}\hfill \\ & & -\mathit{the}-\mathit{SuperHyperVertices}\hfill \\ & & -\mathit{of}-\mathit{neutrosophic}-\mathit{cardinality}-\mathit{of}-\mathit{first}-\mathit{SuperHyperPart}-\mathit{multiplies}-\hfill \\ & & \mathit{sec}\mathit{ond}-\mathit{one}-\mathit{plus}-\mathit{plus}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\mathit{SuperHyperVertices}|min|\hfill \\ & & \mathit{the}\mathit{SuperHyperSets}\mathit{of}\mathit{the}\hfill \\ & & \mathit{interior}\mathit{neutrosophic}\hfill \\ & & \mathit{SuperHyperVertices},\hfill \\ & & \mathit{with}\mathit{only}\mathit{all}\hfill \\ & & \mathit{exceptions}\mathit{in}\mathit{the}\mathit{form}\hfill \\ & & \mathit{of}\mathit{SuperHyperNeighbors}\hfill \\ & & \mathit{excluding}\mathit{one},\hfill \\ & & \mathit{from}\mathit{same}\hfill \\ & & \mathit{SuperHyperEdge}.\hfill \\ & & {|}_{\mathit{neutrosophic}}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Proof. 

Assume a connected neutrosophic SuperHyperBipartite $NSHB:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperBipartite $NSHB:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only all exceptions in the form of interior neutrosophic SuperHyperVertices titled neutrosophic SuperHyperNeighbors with only one exception. a neutrosophic Failed SuperHyperStable has the number of the neutrosophic cardinality of the first SuperHyperPart multiplies with the neutrosophic cardinality of the second SuperHyperPart plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathrm{The}\mathrm{number}-\mathrm{of}-\mathrm{all}\hfill \\ & & -\mathrm{the}-\mathrm{SuperHyperVertices}\hfill \\ & & -\mathrm{of}-\mathrm{neutrosophic}-\mathrm{cardinality}-\mathrm{of}-\mathrm{first}-\mathrm{SuperHyperPart}-\mathrm{multiplies}-\hfill \\ & & \sec \mathrm{ond}-\mathrm{one}-\mathrm{plus}-\mathrm{plus}\hfill \\ & & \mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\mathrm{SuperHyperVertices}|min|\hfill \\ & & \mathrm{the}\mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\hfill \\ & & \mathrm{interior}\mathrm{neutrosophic}\hfill \\ & & \mathrm{SuperHyperVertices},\hfill \\ & & \mathrm{with}\mathrm{only}\mathrm{all}\hfill \\ & & \mathrm{exceptions}\mathrm{in}\mathrm{the}\mathrm{form}\hfill \\ & & \mathrm{of}\mathrm{SuperHyperNeighbors}\hfill \\ & & \mathrm{excluding}\mathrm{one},\hfill \\ & & \mathrm{from}\mathrm{same}\hfill \\ & & \mathrm{SuperHyperEdge}.\hfill \\ & & {|}_{\mathrm{neutrosophic}}\mathrm{cardinality}\mathrm{amid}\mathrm{those}\mathrm{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 5. 

In the Figure 24, the connected neutrosophic SuperHyperBipartite $NSHB:(V,E),$ is highlighted and featured.

By using the Figure 24 and the Table 4, the neutrosophic SuperHyperBipartite $NSHB:(V,E),$ is obtained.

The obtained SuperHyperSet, by the Algorithm in previous result, of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperBipartite $NSHB:(V,E),$ in the neutrosophic SuperHyperModel 24,

$$\begin{array}{ccc}& & \{V_1,\{C_4,D_4,E_4,H_4\},\hfill \\ & & \{K_4,J_4,L_4,O_4\},\{W_2,Z_2,C_3\},\{C_{13},Z_{12},V_{12},W_{12}\},\hfill \end{array}$$

is the neutrosophic Failed SuperHyperStable.

Proposition 55. 

Assume a connected neutrosophic SuperHyperMultipartite $NSHM:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only one exception in the form of interior neutrosophic SuperHyperVertices from a SuperHyperPart and only one exception in the form of interior neutrosophic SuperHyperVertices from another SuperHyperPart titled “neutrosophic SuperHyperNeighbors ” with neglecting and ignoring one of them. A neutrosophic Failed SuperHyperStable has the number of all the summation on the neutrosophic cardinality of the all SuperHyperParts form distinct neutrosophic SuperHyperEdges plus one. Thus,

$$\begin{array}{ccc}& & \mathit{N}\mathit{e}\mathit{u}\mathit{t}\mathit{r}\mathit{o}\mathit{s}\mathit{o}\mathit{p}\mathit{h}\mathit{i}\mathit{c}\mathit{n}\mathit{e}\mathit{u}\mathit{t}\mathit{r}\mathit{o}\mathit{s}\mathit{o}\mathit{p}\mathit{h}\mathit{i}\mathit{c}\mathit{F}\mathit{a}\mathit{i}\mathit{l}\mathit{e}\mathit{d}\mathit{S}\mathit{u}\mathit{p}\mathit{e}\mathit{r}\mathit{H}\mathit{y}\mathit{p}\mathit{e}\mathit{r}\mathit{S}\mathit{t}\mathit{a}\mathit{b}\mathit{l}\mathit{e}=\{\mathit{the}-\mathit{number}-\mathit{of}-\mathit{all}-\mathit{the}-\hfill \\ & & \mathit{summation}-\mathit{on}-\mathit{the}-\hfill \\ & & \mathit{neutrosophic}-\hfill \\ & & \mathit{cardinality}-\mathit{of}-\mathit{the}-\mathit{all}-\hfill \\ & & \mathit{SuperHyperParts}-\mathit{form}-\hfill \\ & & \mathit{distinct}-\mathit{neutrosophic}-\hfill \\ & & \mathit{SuperHyperEdges}-\hfill \\ & & \mathit{plus}-\mathit{one}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\mathit{SuperHyperVertices}|min|\mathit{of}\mathit{the}\mathit{interior}\hfill \\ & & \mathit{neutrosophic}\hfill \\ & & \mathit{SuperHyperVertices}\hfill \\ & & \mathit{with}\mathit{only}\mathit{one}\hfill \\ & & \mathit{exception}\mathit{in}\mathit{the}\hfill \\ & & \mathit{form}\mathit{of}\mathit{interior}\hfill \\ & & \mathit{neutrosophic}\hfill \\ & & \mathit{SuperHyperVertices}\hfill \\ & & \mathit{from}\mathrm{a}\mathit{SuperHyperPart}\hfill \\ & & \mathit{and}\mathit{only}\mathit{one}\hfill \\ & & \mathit{exception}\mathit{in}\mathit{the}\mathit{form}\hfill \\ & & \mathit{of}\mathit{interior}\hfill \\ & & \mathit{neutrosophic}\hfill \\ & & \mathit{SuperHyperVertices}\hfill \\ & & \mathit{from}\mathit{another}\hfill \\ & & \mathit{SuperHyperPart}\mathit{titled}\hfill \\ & & “\mathit{neutrosophic}\hfill \\ & & \mathit{SuperHyperNeighbors}“\mathit{with}\hfill \\ & & \mathit{neglecting}\mathit{and}\hfill \\ & & \mathit{ignoring}\mathit{one}\mathit{of}\mathit{them}.\hfill \\ & & {|}_{\mathit{neutrosophic}}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Proof. 

Assume a connected neutrosophic SuperHyperMultipartite $NSHM:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperMultipartite $NSHM:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices with only one exception in the form of interior neutrosophic SuperHyperVertices from a SuperHyperPart and only one exception in the form of interior neutrosophic SuperHyperVertices from another SuperHyperPart titled “neutrosophic SuperHyperNeighbors ” with neglecting and ignoring one of them. a neutrosophic Failed SuperHyperStable has the number of all the summation on the neutrosophic cardinality of the all SuperHyperParts form distinct neutrosophic SuperHyperEdges plus one. Thus,

$$\begin{array}{ccc}& & NeutrosophicneutrosophicFailedSuperHyperStable=\{\mathrm{the}-\mathrm{number}-\mathrm{of}-\mathrm{all}-\mathrm{the}-\hfill \\ & & \mathrm{summation}-\mathrm{on}-\mathrm{the}-\hfill \\ & & \mathrm{neutrosophic}-\hfill \\ & & \mathrm{cardinality}-\mathrm{of}-\mathrm{the}-\mathrm{all}-\hfill \\ & & \mathrm{SuperHyperParts}-\mathrm{form}-\hfill \\ & & \mathrm{distinct}-\mathrm{neutrosophic}-\hfill \\ & & \mathrm{SuperHyperEdges}-\hfill \\ & & \mathrm{plus}-\mathrm{one}\hfill \\ & & \mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\mathrm{SuperHyperVertices}|min|\mathrm{of}\mathrm{the}\mathrm{interior}\hfill \\ & & \mathrm{neutrosophic}\hfill \\ & & \mathrm{SuperHyperVertices}\hfill \\ & & \mathrm{with}\mathrm{only}\mathrm{one}\hfill \\ & & \mathrm{exception}\mathrm{in}\mathrm{the}\hfill \\ & & \mathrm{form}\mathrm{of}\mathrm{interior}\hfill \\ & & \mathrm{neutrosophic}\hfill \\ & & \mathrm{SuperHyperVertices}\hfill \\ & & \mathrm{from}\mathrm{a}\mathrm{SuperHyperPart}\hfill \\ & & \mathrm{and}\mathrm{only}\mathrm{one}\hfill \\ & & \mathrm{exception}\mathrm{in}\mathrm{the}\mathrm{form}\hfill \\ & & \mathrm{of}\mathrm{interior}\hfill \\ & & \mathrm{neutrosophic}\hfill \\ & & \mathrm{SuperHyperVertices}\hfill \\ & & \mathrm{from}\mathrm{another}\hfill \\ & & \mathrm{SuperHyperPart}\mathrm{titled}\hfill \\ & & ”\mathrm{neutrosophic}\hfill \\ & & \mathrm{SuperHyperNeighbors}”\mathrm{with}\hfill \\ & & \mathrm{neglecting}\mathrm{and}\hfill \\ & & \mathrm{ignoring}\mathrm{one}\mathrm{of}\mathrm{them}.\hfill \\ & & {|}_{\mathrm{neutrosophic}}\mathrm{cardinality}\mathrm{amid}\mathrm{those}\mathrm{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 6. 

In the Figure 25, the connected neutrosophic SuperHyperMultipartite $NSHM:(V,E),$ is highlighted and featured. By using the Figure 25 and the Table 5, the neutrosophic SuperHyperMultipartite $NSHM:(V,E),$ is obtained.

The obtained SuperHyperSet, by the Algorithm in previous result, of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperMultipartite $NSHM:(V,E),$

$$\begin{array}{ccc}& & \left\{\right\{\{L_4,E_4,O_4,D_4,J_4,K_4,H_4\},\hfill \\ & & \{S_{10},R_{10},P_{10}\},\hfill \\ & & \{Z_7,W_7\},\{U_7,V_7\}\},\hfill \end{array}$$

in the neutrosophic SuperHyperModel 25, is the neutrosophic Failed SuperHyperStable.

Proposition 56. 

Assume a connected neutrosophic SuperHyperWheel $NSHW:(V,E).$ Then a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices, excluding the SuperHyperCenter, with only one exception in the form of interior neutrosophic SuperHyperVertices from same neutrosophic SuperHyperEdge with the exclusion once. A neutrosophic Failed SuperHyperStable has the number of all the number of all the neutrosophic SuperHyperEdges have no common neutrosophic SuperHyperNeighbors for a neutrosophic SuperHyperVertex with the exclusion once. Thus,

$$\begin{array}{ccc}& & NeutrosophicFailedSuperHyperStable=\hfill \\ & & \{\mathit{The}-\mathit{number}-\hfill \\ & & \mathit{of}-\mathit{all}-\mathit{the}-\mathit{number}-\mathit{of}-\mathit{all}-\mathit{the}-\mathit{neutrosophic}-\mathit{SuperHyperEdges}-\hfill \\ & & \mathit{have}-\mathit{no}-\mathit{common}-\mathit{neutrosophic}-\mathit{SuperHyperNeighbors}-\mathit{for}-\mathrm{a}-\hfill \\ & & \mathit{neutrosophic}-\mathit{SuperHyperVertex}-\mathit{with}-\mathit{the}-\mathit{exclusion}-\mathit{once}\hfill \\ & & \mathit{SuperHyperSets}\mathit{of}\mathit{the}\mathit{SuperHyperVertices}|min|\mathit{SuperHyperSet}\hfill \\ & & \mathit{of}\mathit{the}\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices},\mathit{excluding}\mathit{the}\hfill \\ & & \mathit{SuperHyperCenter},\mathit{with}\mathit{only}\mathit{one}\mathit{exception}\mathit{in}\hfill \\ & & \mathit{the}\mathit{form}\mathit{of}\mathit{interior}\mathit{neutrosophic}\mathit{SuperHyperVertices}\mathit{from}\hfill \\ & & \mathit{same}\mathit{neutrosophic}\mathit{SuperHyperEdge}\mathit{with}\hfill \\ & & {\mathit{the}\mathit{exclusion}\mathit{once}.|}_{\mathit{neutrosophic}}\mathit{cardinality}\mathit{amid}\mathit{those}\mathit{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

Proof. 

Assume a connected neutrosophic SuperHyperWheel $NSHW:(V,E).$ Let a neutrosophic SuperHyperEdge has some neutrosophic SuperHyperVertices. Consider all numbers of those neutrosophic SuperHyperVertices from that neutrosophic SuperHyperEdge excluding more than two distinct neutrosophic SuperHyperVertices, exclude to any given SuperHyperSet of the neutrosophic SuperHyperVertices. Consider there’s a neutrosophic Failed SuperHyperStable with the least neutrosophic cardinality, the lower sharp bound for neutrosophic cardinality. Assume a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \left\{z\right\}$ is a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t have the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. The SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,y,z\}$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices but it isn’t a neutrosophic Failed SuperHyperStable . Since it doesn’t do the procedure such that such that there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. [there’er at least three neutrosophic SuperHyperVertices inside implying there’s, sometimes in the connected neutrosophic SuperHyperGraph $NSHG:(V,E),$ a neutrosophic SuperHyperVertex, titled its neutrosophic SuperHyperNeighbor, to that neutrosophic SuperHyperVertex in the SuperHyperSet S so as S doesn’t do “the procedure”.]. There’re only two neutrosophic SuperHyperVertices inside the intended SuperHyperSet, $V\setminus V\setminus \{x,z\}.$ Thus the obvious neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$ is up. The obvious simple type-SuperHyperSet of the neutrosophic Failed SuperHyperStable, $V\setminus V\setminus \{x,z\},$is a SuperHyperSet, $V\setminus V\setminus \{x,z\},$includes only two neutrosophic SuperHyperVertices doesn’t form any kind of pairs are titled neutrosophic SuperHyperNeighbors in a connected neutrosophic SuperHyperGraph $NSHG:(V,E).$ Since the SuperHyperSet of the neutrosophic SuperHyperVertices $V\setminus V\setminus \{x,z\},$ is the maximum neutrosophic cardinality of a SuperHyperSet S of neutrosophic SuperHyperVertices such that$V\left(G\right)$ there’s a neutrosophic SuperHyperVertex to have a neutrosophic SuperHyperEdge in common. Thus, in a connected neutrosophic SuperHyperWheel $NSHW:(V,E),$ a neutrosophic Failed SuperHyperStable is a SuperHyperSet of the interior neutrosophic SuperHyperVertices, excluding the SuperHyperCenter, with only one exception in the form of interior neutrosophic SuperHyperVertices from same neutrosophic SuperHyperEdge with the exclusion once. a neutrosophic Failed SuperHyperStable has the number of all the number of all the neutrosophic SuperHyperEdges have no common neutrosophic SuperHyperNeighbors for a neutrosophic SuperHyperVertex with the exclusion once. Thus,

$$\begin{array}{ccc}& & NeutrosophicFailedSuperHyperStable=\hfill \\ & & \{\mathrm{The}-\mathrm{number}-\hfill \\ & & \mathrm{of}-\mathrm{all}-\mathrm{the}-\mathrm{number}-\mathrm{of}-\mathrm{all}-\mathrm{the}-\mathrm{neutrosophic}-\mathrm{SuperHyperEdges}-\hfill \\ & & \mathrm{have}-\mathrm{no}-\mathrm{common}-\mathrm{neutrosophic}-\mathrm{SuperHyperNeighbors}-\mathrm{for}-\mathrm{a}-\hfill \\ & & \mathrm{neutrosophic}-\mathrm{SuperHyperVertex}-\mathrm{with}-\mathrm{the}-\mathrm{exclusion}-\mathrm{once}\hfill \\ & & \mathrm{SuperHyperSets}\mathrm{of}\mathrm{the}\mathrm{SuperHyperVertices}|min|\mathrm{SuperHyperSet}\hfill \\ & & \mathrm{of}\mathrm{the}\mathrm{interior}\mathrm{neutrosophic}\mathrm{SuperHyperVertices},\mathrm{excluding}\mathrm{the}\hfill \\ & & \mathrm{SuperHyperCenter},\mathrm{with}\mathrm{only}\mathrm{one}\mathrm{exception}\mathrm{in}\hfill \\ & & \mathrm{the}\mathrm{form}\mathrm{of}\mathrm{interior}\mathrm{neutrosophic}\mathrm{SuperHyperVertices}\mathrm{from}\hfill \\ & & \mathrm{same}\mathrm{neutrosophic}\mathrm{SuperHyperEdge}\mathrm{with}\hfill \\ & & {\mathrm{the}\mathrm{exclusion}\mathrm{once}.|}_{\mathrm{neutrosophic}}\mathrm{cardinality}\mathrm{amid}\mathrm{those}\mathrm{SuperHyperSets}.\}\hfill \end{array}$$

Where ${\sigma }_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. □

Example 7. 

In the Figure 26, the connected neutrosophic SuperHyperWheel $NSHW:(V,E),$ is highlighted and featured.

By using the Figure 26 and the Table 6, the neutrosophic SuperHyperWheel $NSHW:(V,E),$ is obtained.

The obtained SuperHyperSet, by the Algorithm in previous result, of the neutrosophic SuperHyperVertices of the connected neutrosophic SuperHyperWheel $NSHW:(V,E),$

$$\begin{array}{ccc}& & \{V_5,\hfill \\ & & \{Z_{13},W_{13},U_{13},V_{13},O_{14}\},\hfill \\ & & \{T_{10},K_{10},J_{10}\},\hfill \\ & & \{E_7,C_7,Z_6\},\{K_7,J_7,L_7\},\hfill \\ & & \{T_{14},U_{14},R_{15},S_{15}\}\},\hfill \end{array}$$

in the neutrosophic SuperHyperModel 26, is the neutrosophic Failed SuperHyperStable.

7. Open Neutrosophic Problems

In what follows, some “Neutrosophic problems” and some “Neutrosophic questions” are Neutrosophicly proposed.

The Failed SuperHyperStable and the neutrosophic Failed SuperHyperStable are Neutrosophicly defined on a real-world Neutrosophic application, titled “Cancer’s neutrosophic recognitions”.

Figure 26. A neutrosophic SuperHyperWheel Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 7

Figure 26. A neutrosophic SuperHyperWheel Associated to the Notions of neutrosophic Failed SuperHyperStable in the Example 7

Question 3. 

Which the else neutrosophic SuperHyperModels could be defined based on Cancer’s neutrosophic recognitions?

Question 4. 

Are there some neutrosophic SuperHyperNotions related to Failed SuperHyperStable and the neutrosophic Failed SuperHyperStable?

Question 5. 

Are there some Neutrosophic Algorithms to be defined on the neutrosophic SuperHyperModels to compute them Neutrosophicly?

Question 6. 

Which the neutrosophic SuperHyperNotions are related to beyond the Failed SuperHyperStable and the neutrosophic Failed SuperHyperStable?

Problem 7. 

The Failed SuperHyperStable and the neutrosophic Failed SuperHyperStable do Neutrosophicly a neutrosophic SuperHyperModel for the Cancer’s neutrosophic recognitions and they’re based Neutrosophicly on neutrosophic Failed SuperHyperStable, are there else Neutrosophicly?

Problem 8. 

Which the fundamental Neutrosophic SuperHyperNumbers are related to these Neutrosophic SuperHyperNumbers types-results?

Problem 9. 

What’s the independent research based on Cancer’s neutrosophic recognitions concerning the multiple types of neutrosophic SuperHyperNotions?

8. Neutrosophic Conclusion and Closing Remarks

In this research, the cancer is chosen as a Neutrosophic phenomenon. Some Neutrosophic general approaches are Neutrosophicly applied on it. Beyond that, some general Neutrosophic arrangements of the Neutrosophic situations are Neutrosophicly redefined alongside detailed-oriented illustrations, clarifications, analysis on the featured dense Neutrosophic figures. The research proposes theoretical Neutrosophic results on the cancer and mentioned Neutrosophic cases only give us the Neutrosophic perspective on the theoretical Neutrosophic aspect with enriched Neutrosophic background of the the mathematical Neutrosophic framework arise from Extreme Failed SuperHyperClique theory, Neutrosophic Failed SuperHyperClique theory, and (Neutrosophic) SuperHyperGraphs theory. In the Table 7, the literatures of this research on what’s happened and what will happen are pointed out and figured out.

Table 7. The Literatures of This Research On What’s Happened and What will Happen

Table 7. The Literatures of This Research On What’s Happened and What will Happen