Recognition of Cancer With Bipartite and Path as Neutrosophic SuperHyperGraph By the Tool Called Stability

1. Scientific Research

Referred to Ref. [2].

Theorem 1.1. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime }}F\left(V_i\right)\left)\right|\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{NSHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\forall V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & T_V^{\prime }(V_i,V_j\in E_{i^{\prime }})\neg \le min{[T_V\left(V_i^{\prime }\right),T_V\left(V_j^{\prime }\right)]}_{V_i^{\prime },V_j^{\prime }\in E_{NSHG}},\hfill \\ & & I_V^{\prime }(V_i,V_j\in E_{i^{\prime }})\neg \le min{[I_V\left(V_i^{\prime }\right),I_V\left(V_j^{\prime }\right)]}_{V_i^{\prime },V_j^{\prime }\in E_{NSHG}},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_{i^{\prime }}\right)(V_i,V_j\in E_{i^{\prime }})\neg \le min{[F_V\left(V_i^{\prime }\right),F_V\left(V_j^{\prime }\right)]}_{V_i^{\prime },V_j^{\prime }\in E_{NSHG}}\hfill \\ & & \}\hfill \end{array}$$

Example 1.2. 

Referred to the Figure 1.

Theorem 1.3. 

Neutrosophic SuperHyperBipartiteStyles-I don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\left\{V_1\right\}}T\left(V_i\right),\sum _{V_i\in V^{\prime }=\left\{V_1\right\}}I\left(V_i\right),\sum _{V_i\in V^{\prime }=\left\{V_1\right\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(0.26,0.26,0.26).\hfill \end{array}$$

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Example 1.4. 

Referred to the Figure 2.

Theorem 1.5. 

Neutrosophic SuperHyperBipartiteStyles-II don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\{V_1,V_2\}}T\left(V_i\right),\sum _{V_i\in V^{\prime }=\{V_1,V_2\}}I\left(V_i\right),\sum _{V_i\in V^{\prime }=\{V_1,V_2\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(0.5,0.5,0.5)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_1\in V^{\prime \prime },\forall V_2\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_1\right),T_{V^{\prime }}\left(V_2\right)]}_{V_1,V_2\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_1\right),I_{V^{\prime }}\left(V_2\right)]}_{V_1,V_2\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_1\right),F_{V^{\prime }}\left(V_2\right)]}_{V_1,V_2\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.6. 

Referred to the Figure 3.

Theorem 1.7. 

Neutrosophic SuperHyperBipartiteStyles-III don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^3}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^3}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^3}}F\left(V_i\right)\left)\right|\hfill \\ & & =(0.75,0.75,0.75)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.8. 

Referred to the Figure 4.

Theorem 1.9. 

Neutrosophic SuperHyperBipartiteStyles-IV don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^4}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^4}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^4}}F\left(V_i\right)\left)\right|\hfill \\ & & =(0.99,0.99,0.99)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.10. 

Referred to the Figure 5.

Theorem 1.11. 

Neutrosophic SuperHyperBipartiteStyles-V don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^5}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^5}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^5}}F\left(V_i\right)\left)\right|\hfill \\ & & =(1.23,1.23,1.23)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.12. 

Referred to the Figure 6.

Theorem 1.13. 

Neutrosophic SuperHyperBipartiteStyles-VI don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^6}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^6}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^6}}F\left(V_i\right)\left)\right|\hfill \\ & & =(1.47,1.47,1.47)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.14. 

Referred to the Figure 7.

Theorem 1.15. 

Neutrosophic SuperHyperBipartiteStyles-VII don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^7}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^7}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }={\left\{V_i\right\}}_{i=1^7}}F\left(V_i\right)\left)\right|\hfill \\ & & =(1.71,1.71,1.71)=\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_i\in V^{\prime \prime },\forall V_j\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.16. 

Referred to the Figure 8.

Theorem 1.17. 

Neutrosophic SuperHyperPathStyles-I don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\{V_1,V_8,V_{12},V_{16},V_{20},V_{25},V_{32}\}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\{V_1,V_8,V_{12},V_{16},V_{20},V_{25},V_{32}\}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\{V_1,V_8,V_{12},V_{16},V_{20},V_{25},V_{32}\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(1.82,1.82,1.82)\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_{i=1,8,12,16,20,25,32}\in V^{\prime \prime },\forall V_{j=1,8,12,16,20,25,32}\in V^{\prime \prime }:\hfill \\ & & T_V^{\prime }(E_k{\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & I_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k}\hfill \\ & & \}\hfill \end{array}$$

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Example 1.18. 

Referred to the Figure 9.

Theorem 1.19. 

Neutrosophic SuperHyperPathStyles-II don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\left\{V_{1-2,8,11-12,15-16,19-20,24-25,32-34}\right\}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-2,8,11-12,15-16,19-20,24-25,32-34}\right\}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-2,8,11-12,15-16,19-20,24-25,32-34}\right\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(3.48,3.48,3.48)\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_{i=1,8,12,16,20,25,32}\in V^{\prime \prime },\forall V_{j=1,8,12,16,20,25,32}\in V^{\prime \prime }:\hfill \\ & & T_V^{\prime }(E_k{\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & I_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & V_{i=1-2,11,15,19,24}\in V^{\prime \prime },\forall V_{j=1-2,11,15,19,24}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=32-34}\in V^{\prime \prime },\forall V_{j=32-34}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

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Example 1.20. 

Referred to the Figure 10.

Theorem 1.21. 

Neutrosophic SuperHyperPathStyles-III don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\left\{V_{1-2,8-12,15-16,19-20,24-25,31-34}\right\}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-2,8-12,15-16,19-20,24-25,31-34}\right\}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-2,8-12,15-16,19-20,24-25,31-34}\right\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(4.2,4.2,4.2)\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_{i=1,8,12,16,20,25,32}\in V^{\prime \prime },\forall V_{j=1,8,12,16,20,25,32}\in V^{\prime \prime }:\hfill \\ & & T_V^{\prime }(E_k{\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & I_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & V_{i=1-2,11,15,19,24}\in V^{\prime \prime },\forall V_{j=1-2,11,15,19,24}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=32-34}\in V^{\prime \prime },\forall V_{j=32-34}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=8-10,31,2}\in V^{\prime \prime },\forall V_{j=8-10,31,2}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

□

Example 1.22. 

Referred to the Figure 11.

Theorem 1.23. 

Neutrosophic SuperHyperPathStyles-IV don’t coincide.

Proof. 

$$\begin{array}{ccc}& & \left|\right(\sum _{V_i\in V^{\prime }=\left\{V_{1-4,8-12-16,19-20,24-25,31-34}\right\}}T\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-4,8-12-16,19-20,24-25,31-34}\right\}}I\left(V_i\right),\hfill \\ & & \sum _{V_i\in V^{\prime }=\left\{V_{1-4,8-12-16,19-20,24-25,31-34}\right\}}F\left(V_i\right)\left)\right|\hfill \\ & & =(5.16,5.16,5.16)\hfill \\ & & =\underset{V^{\prime \prime }\subseteq V_{SHG}}{max}\left|\right\{(\sum _{V_i\in V^{\prime \prime }}T\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}I\left(V_i\right),\sum _{V_i\in V^{\prime \prime }}F\left(V_i\right))|\hfill \\ & & V_{i=1,8,12,16,20,25,32}\in V^{\prime \prime },\forall V_{j=1,8,12,16,20,25,32}\in V^{\prime \prime }:\hfill \\ & & T_V^{\prime }(E_k{\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & I_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_k\right){\sim }^{\mathrm{NOT}\mathrm{DEFINED}}min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\notin E_k},\hfill \\ & & V_{i=1-2,11,15,19,24}\in V^{\prime \prime },\forall V_{j=1-2,11,15,19,24}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=32-34}\in V^{\prime \prime },\forall V_{j=32-34}\in V^{\prime \prime }:\hfill \\ & \end{array}$$

$$\begin{array}{c}\\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=8-10,31,2}\in V^{\prime \prime },\forall V_{j=8-10,31,2}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=12-14,3-4}\in V^{\prime \prime },\forall V_{j=12-14,3-4}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & V_{i=4,16}\in V^{\prime \prime },\forall V_{j=4,16}\in V^{\prime \prime }:\hfill \\ & & \\ & & T_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[T_{V^{\prime }}\left(V_i\right),T_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & I_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[I_{V^{\prime }}\left(V_i\right),I_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \mathrm{and}{F}_V^{\prime }\left(E_1\right)=(0.26,0.26,0.26)\neg \le (0.24,0.24,0.24)=\hfill \\ & & min{[F_{V^{\prime }}\left(V_i\right),F_{V^{\prime }}\left(V_j\right)]}_{V_i,V_j\in E_1},\hfill \\ & & \}\hfill \end{array}$$

□

Example 1.24. 

Referred to the Figure 12.