Power Graphs as Hypergraphs and n^th Power Graphs as n-SuperHyperGraphs

1. Preliminaries

We begin by reviewing the basic terminology and notation used throughout this paper. Unless specified otherwise, all graphs are assumed to be undirected, finite, and simple. For more extensive discussions of particular operations and concepts, the reader is referred to the literature.

1.1. Power Graph and Directed Power Graph

A Power Graph has group elements as vertices, with edge joining elements when one is a power of the other. Directed Power Graph uses group elements as vertices and a directed edge $x\to y$ whenever $y=x^m$ for some $m\in \mathbb{N}$.

Definition 1.1 (Group).

[6,7,8] A group is a pair $(G,·)$ consisting of a nonempty set G and a binary operation

$$·:G\times G⟶G$$

satisfying the following axioms:

1.

Associativity: For all $a,b,c\in G$,

$$(a·b)·c=a·(b·c).$$

2.

Identity: There exists a unique element $e\in G$ such that for all $a\in G$,

$$e·a=a\mathrm{and}a·e=a.$$

3.

Inverse: For each $a\in G$, there exists an element $a^{-1}\in G$ such that

$$a·a^{-1}=e\mathrm{and}{a}^{-1}·a=e.$$

Definition 1.2 (Power Graph).

[9,10,11] Let G be a group. The power graph of G is the (undirected) graph

$$P\left(G\right)=(V,E),V=G,$$

whose edge set is

$$E=\left\{\{x,y\}\subseteq G|x\ne y,\exists m,n\in \mathbb{N}:x^m=y\mathrm{or}{y}^n=x\right\}.$$

Equivalently, $P\left(G\right)$ is the underlying simple graph of $\overrightarrow{P}\left(G\right)$.

Example 1.3 (Power Graph of the Cyclic Group of Order 4).

Let $G=\langle a\mid a^4=e\rangle =\{e,a,a^2,a^3\}$. Its power graph $P\left(G\right)=(V,E)$ is given by

$$V=\{e,a,a^2,a^3\},E=\left\{\{e,a\},\{e,a^2\},\{e,a^3\},\{a,a^2\},\{a,a^3\},\{a^2,a^3\}\right\}.$$

Definition 1.4 (Directed Power Graph).

[9,11] Let G be a group. The directed power graph of G is the digraph

$$\overrightarrow{P}\left(G\right)=(V,A),V=G,$$

where

$$A=\left\{(x,y)\in G\times G|x\ne y,y=x^m\mathrm{for}\mathrm{some}m\in \mathbb{N}\right\}.$$

Example 1.5 (Directed Power Graph of the Cyclic Group of Order 4).

On the same group G, the directed power graph $\overrightarrow{P}\left(G\right)=(V,A)$ has

$$V=\{e,a,a^2,a^3\},A=\left\{(a,a^2),(a,a^3),(a,e),(a^2,e),(a^3,a^2),(a^3,a),(a^3,e)\right\}.$$

Here each arc $(x,y)$ satisfies $y=x^m$ for some $m\in \mathbb{N}$ and $x\ne y$.

1.2. SuperHyperGraph

A hypergraph generalizes a standard graph by allowing hyperedges that can join any number of vertices simultaneously [3,12,13,14]. Extending this idea, a SuperHyperGraph incorporates iterated powerset constructions to capture hierarchical relationships among hyperedges, a topic of growing interest in recent studies [5,15,16,17,18,19]. Practical applications of SuperHyperGraphs include molecular modeling, network analysis, and signal processing [20,21,22,23,24,25]. In what follows, the integer parameter n in the nth powerset and in an n-SuperHyperGraph always denotes a nonnegative integer.

Definition 1.6 (Base Set).

A base set S is the underlying domain from which all further constructions are drawn. Formally,

$$S=\{x\mid x\mathrm{belongs}\mathrm{to}\mathrm{the}\mathrm{specified}\mathrm{universe}\}.$$

Every element appearing in $\mathcal{P}\left(S\right)$ or in iterated powersets ${\mathcal{P}}_n\left(S\right)$ is an element of S.

Definition 1.7 (Powerset).

The powerset of a set S, written $\mathcal{P}\left(S\right)$, is the collection of all subsets of S, including ∅ and S itself:

$$\mathcal{P}\left(S\right)=\{A\mid A\subseteq S\}.$$

Definition 1.8 (Hypergraph).

[3,26] A hypergraph $H=\left(V\right(H),E(H\left)\right)$ consists of

• A finite vertex set $V\left(H\right)$.
• A finite collection $E\left(H\right)$ of nonempty subsets of $V\left(H\right)$, called hyperedges.

Hypergraphs are well suited to model higher-order interactions among elements of $V\left(H\right)$.

Theorem 1.9. 

Let G be a group and let

$$P\left(G\right)=(V,E)\mathrm{with}V=G,E=\left\{\{x,y\}\subseteq G\mid x\ne y,\exists m,n\in \mathbb{N}:x^m=y\mathrm{or}{y}^n=x\right\}$$

be its power graph (Definition 1.2). Then

$$H=\left(G,E\right)$$

is a 2-uniform hypergraph (Definition 1.8), and its 2-section (the simple graph obtained by replacing each hyperedge by an undirected edge) is exactly $P\left(G\right)$.

Proof. 

We verify that $H=(G,E)$ satisfies the hypergraph axioms:

• $V\left(H\right)=G$ is finite (since G is finite or else this construction still formally applies).
• Each $e\in E$ is a nonempty subset of G, and by construction $\left|e\right|=2$.

Hence H is a 2-uniform hypergraph. Its 2-section is formed by interpreting each hyperedge $\{x,y\}$ as an undirected edge between x and y. But exactly those pairs $\{x,y\}$ appear in E for which $x^m=y$ or $y^n=x$, so the resulting simple graph coincides with the power graph $P\left(G\right)$. □

Definition 1.10 (n-th Powerset).

[27,28,29,30,31] The n-th powerset of a set X, denoted $P_n\left(X\right)$, is defined by:

$$P_1\left(X\right)=\mathcal{P}\left(X\right),P_{n+1}\left(X\right)=\mathcal{P}\left(P_n\left(X\right)\right),n\ge 1.$$

The corresponding nonempty powerset $P_n^{*}\left(X\right)$ is obtained by iterating ${\mathcal{P}}^{*}(·)$, where ${\mathcal{P}}^{*}\left(Y\right)=\mathcal{P}\left(Y\right)\setminus \{Ø\}$.

Definition 1.11 (n-SuperHyperGraph).

[32,33,34] Let $V_0$ be a finite base set. Define iteratively

$${\mathcal{P}}^0\left(V_0\right)=V_0,{\mathcal{P}}^{k+1}\left(V_0\right)=\mathcal{P}\left({\mathcal{P}}^k\left(V_0\right)\right).$$

An n-SuperHyperGraph is a pair

$${\mathrm{SuHyG}}^{\left(n\right)}=(V,E),V,E\subseteq {\mathcal{P}}^n\left(V_0\right),$$

where each element of V is called an n-supervertex and each element of E an n-superedge.

Example 1.12 (3-SuperHyperGraph: Global Project Organization).

We illustrate a 3-SuperHyperGraph by modeling the hierarchy of a multinational corporation’s project structure.

Base set: employees.

$$V_0=\{\mathrm{Alice},\mathrm{Bob},\mathrm{Carol},\mathrm{Dave}\}.$$

First–level (teams). Elements of ${\mathcal{P}}^1\left(V_0\right)$:

$$T_1=\{\mathrm{Alice},\mathrm{Bob}\},T_2=\{\mathrm{Bob},\mathrm{Carol}\},T_3=\{\mathrm{Carol},\mathrm{Dave}\}.$$

Second–level (departments). Elements of ${\mathcal{P}}^2\left(V_0\right)=\mathcal{P}\left({\mathcal{P}}^1\left(V_0\right)\right)$. Select:

$$D_1=\{T_1,T_2\},D_2=\{T_2,T_3\}.$$

Third–level (global divisions). Elements of ${\mathcal{P}}^3\left(V_0\right)$. Choose two representative supervertices:

$$v_1=\{D_1,D_2\},v_2=\{D_1,\{T_1,T_3\}\}.$$

Thus we set

$$V=\{v_1,v_2\}\subseteq {\mathcal{P}}^3\left(V_0\right),E=\left\{\{v_1,v_2\}\right\}\subseteq \mathcal{P}\left(V\right).$$

Here $v_1$ models the “Asia–Europe Division” (linking Department 1 and Department 2), while $v_2$ models the “Global Integration Division” (combining Department 1 with the cross-team set $\{T_1,T_3\}$).

Interpretation. The 3-supervertices $v_1,v_2$ represent high–level divisions composed of lower–level departments and teams. The single superedge $e=\{v_1,v_2\}$ indicates a strategic collaboration between these two global divisions on a company-wide initiative.

Hence

$${\mathrm{SuHyG}}^{\left(3\right)}=\left(V,E\right)$$

is a concrete instance of a 3-SuperHyperGraph reflecting a real-world organizational hierarchy.

1.3. Directed SuperHyperGraph

Directed SuperHyperGraphs are graph classes that extend SuperHyperGraphs, respectively, in a manner analogous to Directed Graphs. Below, we present their formal definitions and illustrative examples.

Definition 1.13 (Directed Hypergraph).

A directed hypergraph is a pair $H=(V,E)$ where

• V is a finite set of vertices, and
• E is a set of directed hyperedges.

Each hyperedge $e\in E$ is an ordered pair

$$e=\left(T_e,H_e\right),$$

where

$$T_e\subseteq V(\mathrm{the}\mathrm{tail}\mathrm{of}e),H_e\subseteq V(\mathrm{the}\mathrm{head}\mathrm{of}e),$$

and one typically requires $T_e\ne Ø$ and $H_e\ne Ø$. This structure generalizes a directed graph by allowing each hyperedge to connect multiple source vertices $T_e$ to multiple target vertices $H_e$ simultaneously.

Theorem 1.14. 

Let G be a group and let $\overrightarrow{P}\left(G\right)=(G,A)$ be its directed power graph (Definition 1.4), where

$$A=\{(x,y)\in G\times G\mid y=x^m\mathrm{for}\mathrm{some}m\in \mathbb{N},x\ne y\}.$$

Define a directed hypergraph

$$H=(V,E_H),V=G,E_H=\left\{\left(\left\{x\right\},\left\{y\right\}\right)\mid (x,y)\in A\right\}.$$

Then H is a directed hypergraph, and its underlying digraph is exactly $\overrightarrow{P}\left(G\right)$. Hence $\overrightarrow{P}\left(G\right)$ is realized as a directed hypergraph with all hyperedges of size one.

Proof. 

First, $V=G$ is finite and nonempty. Each hyperedge in $E_H$ is of the form $(T_e,H_e)$ with

$$T_e=\left\{x\right\}\subseteq V,H_e=\left\{y\right\}\subseteq V,$$

and both $T_e$ and $H_e$ are nonempty by construction. Therefore H satisfies the requirements of the Definition.

Next, the underlying directed graph of H has an arc $x\to y$ precisely when there exists a hyperedge $(\left\{x\right\},\left\{y\right\})\in E_H$. By definition of $E_H$, this occurs if and only if $(x,y)\in A$, i.e. $y=x^m$ for some $m\ge 1$. This is exactly the arc-set of $\overrightarrow{P}\left(G\right)$. Hence the two directed graphs coincide. □

Definition 1.15 (Directed n-SuperHyperGraph).

(cf.[22,34,35]) Let S be a nonempty base set and let $n\ge 0$ be an integer. Define iterated powersets by

$${\mathcal{P}}^0\left(S\right)=S,{\mathcal{P}}^{k+1}\left(S\right)=\mathcal{P}\left({\mathcal{P}}^k\left(S\right)\right)(k\ge 0).$$

A directed n-SuperHyperGraph is a pair

$${\mathrm{DSuHG}}^{\left(n\right)}=(V,E),$$

where

$$V\subseteq {\mathcal{P}}^n\left(S\right),E\subseteq {\mathcal{P}}^n\left(S\right)\times {\mathcal{P}}^n\left(S\right),$$

and each directed n-superedge $e\in E$ is an ordered pair

$$e=\left(\mathrm{Tail}\left(e\right),\mathrm{Head}\left(e\right)\right),\mathrm{Tail}\left(e\right),\mathrm{Head}\left(e\right)\subseteq {\mathcal{P}}^n\left(S\right),$$

typically both nonempty. Such an e carries “flow” from the entire set $\mathrm{Tail}\left(e\right)$ of n-supervertices into $\mathrm{Head}\left(e\right)$.

Example 1.16 (Directed 2-SuperHyperGraph: Corporate Workflow).

We model a simplified approval process in a company with two teams and a central review committee.

Base set (employees):

$$S=\{\mathrm{Alice},\mathrm{Bob},\mathrm{Carol},\mathrm{Dave}\}.$$

First–level (teams):

$$T_1=\{\mathrm{Alice},\mathrm{Bob}\},T_2=\{\mathrm{Carol},\mathrm{Dave}\}\subseteq \mathcal{P}\left(S\right).$$

Second–level (departments):

$$D_1=\left\{T_1\right\},D_2=\left\{T_2\right\},D_{\mathrm{all}}=\{T_1,T_2\}\subseteq {\mathcal{P}}^2\left(S\right).$$

Vertex set:

$$V=\{D_1,D_2,D_{\mathrm{all}}\}\subseteq {\mathcal{P}}^2\left(S\right).$$

Directed superedges (approval flow):

$$E=\left\{\left(\left\{D_1\right\},\left\{D_{\mathrm{all}}\right\}\right),\left(\left\{D_2\right\},\left\{D_{\mathrm{all}}\right\}\right)\right\}\subseteq {\mathcal{P}}^2\left(S\right)\times {\mathcal{P}}^2\left(S\right).$$

Interpretation. Each directed superedge $\left(\left\{D_i\right\},\left\{D_{\mathrm{all}}\right\}\right)$ represents the flow of deliverables (or approval requests) from department $D_i$ into the central committee $D_{\mathrm{all}}$. Thus

$${\mathrm{DSuHG}}^{\left(2\right)}=(V,E)$$

is a concrete directed 2-SuperHyperGraph illustrating this corporate workflow.

Theorem 1.17. 

Every directed hypergraph can be realized as a directed 1-SuperHyperGraph. Concretely, if

$$H=(V_0,E),E\subseteq \{(T_e,H_e)\mid T_e,H_e\subseteq V_0,T_e\ne Ø,H_e\ne Ø\},$$

then setting

$$S=V_0,V=\left\{\left\{v\right\}\mid v\in V_0\right\}\subseteq {\mathcal{P}}^1\left(S\right),$$

and for each directed hyperedge $e=(T_e,H_e)\in E$ defining

$$\mathrm{Tail}\left(e\right)=\{\left\{v\right\}\mid v\in T_e\},\mathrm{Head}\left(e\right)=\{\left\{v\right\}\mid v\in H_e\},$$

yields a directed 1-SuperHyperGraph ${\mathrm{DSuHG}}^{\left(1\right)}=(V,E^{\prime })$ with

$$E^{\prime }=\left\{\left(\mathrm{Tail}\right(e),\mathrm{Head}(e\left)\right)\mid e\in E\right\}\subseteq {\mathcal{P}}^1\left(S\right)\times {\mathcal{P}}^1\left(S\right).$$

Proof. 

We verify that $(V,E^{\prime })$ satisfies Definition 1.15 for $n=1$:

• By construction, $V=\{\left\{v\right\}\mid v\in V_0\}\subseteq {\mathcal{P}}^1\left(S\right)$ and is nonempty.
• Each $\mathrm{Tail}\left(e\right)$ and $\mathrm{Head}\left(e\right)$ is a nonempty subset of V, since $T_e,H_e\ne Ø$.
• Thus every $(\mathrm{Tail}\left(e\right),\mathrm{Head}\left(e\right))\in E^{\prime }$ is an element of ${\mathcal{P}}^1\left(S\right)\times {\mathcal{P}}^1\left(S\right)$.

Hence ${\mathrm{DSuHG}}^{\left(1\right)}=(V,E^{\prime })$ is a directed 1-SuperHyperGraph. Since directed hyperedges $(T_e,H_e)$ and directed 1-superedges $\left(\mathrm{Tail}\right(e),\mathrm{Head}(e\left)\right)$ correspond bijectively, H is represented exactly by ${\mathrm{DSuHG}}^{\left(1\right)}$. □

Corollary 1.18. 

Every directed hypergraph is a special case of a directed n-SuperHyperGraph for any $n\ge 1$, via the inclusion ${\mathcal{P}}^1\left(S\right)\subseteq {\mathcal{P}}^n\left(S\right)$.

Proof. 

Embed each singleton $\left\{v\right\}\in {\mathcal{P}}^1\left(S\right)$ into ${\mathcal{P}}^n\left(S\right)$ by iterated singletoning: $\left\{v\right\}\mapsto \left\{\right\{\dots \left\{\right\{v\left\}\right\}\dots \left\}\right\}$. The same construction of tails and heads yields a directed n-SuperHyperGraph isomorphic to the directed 1-case. □

2. Main Results

As the main contributions of this paper, we show that:

• Every n-th power graph can be realized as a special case of an n-SuperHyperGraph.
• Every directed n-th power graph can be realized as a special case of a directed n-SuperHyperGraph.

2.1. n-th Power Graphs

Vertices are n-fold powersets of group G; two vertices are adjacent if one equals the m-th power-set image of the other.

Definition 2.1 (Iterated Exponentiation).

Let G be a group and $m\in \mathbb{N}$. Define maps

$${\phi }_m^{\left(1\right)}:G\to G,x\mapsto x^m,$$

and recursively for $k\ge 1$,

$${\phi }_m^{(k+1)}:{\mathcal{P}}^{k+1}\left(G\right)\to {\mathcal{P}}^{k+1}\left(G\right),A\mapsto \left\{{\phi }_m^{\left(k\right)}\left(a\right)\mid a\in A\right\}.$$

Definition 2.2 (Undirected n-th Power Graph).

Fix $n\ge 1$. Let

$$V_n={\mathcal{P}}^n\left(G\right),E_n=\left\{\{A,B\}\subseteq V_n|A\ne B,B={\phi }_m^{\left(n\right)}\left(A\right)\mathrm{or}A={\phi }_m^{\left(n\right)}\left(B\right)\mathrm{for}\mathrm{some}m\in \mathbb{N}\right\}.$$

Then the n-th power graph is the simple graph

$$P^{\left(n\right)}\left(G\right)=\left(V_n,E_n\right).$$

Example 2.3 (Rotating Shift Patterns as an Undirected 2nd Power Graph).

We model a weekly rotating shift system on a seven-day cycle as an undirected 2nd power graph.

Base group and daily blocks. Let

$$G={\mathbb{Z}}_7=\{0,1,2,3,4,5,6\},$$

with addition modulo 7 representing days of the week. Define two daily shift blocks:

$$A=\{0,1,2\}\left(\mathrm{Mon}\text{–}\mathrm{Wed}\right),B=\{3,4\}\left(\mathrm{Thu}\text{–}\mathrm{Fri}\right).$$

Weekly patterns ($V_2$). Vertices are collections of these blocks in one week:

$$P=\{A,B\},Q=\{A+2,B+2\},R=\{A+3,B+3\},$$

where $A+m=\{x+m(mod7)\mid x\in A\}$. Thus

$$V_2=\{P,Q,R\}\subseteq {\mathcal{P}}^2\left(G\right).$$

Edge relation ($E_2$). In the undirected 2nd power graph $P^{\left(2\right)}\left(G\right)=(V_2,E_2)$, two patterns $\{X,Y\}\subseteq V_2$ are adjacent precisely if one is obtained by applying the same shift m to the other:

$$E_2=\left\{\{P,Q\},\{P,R\},\{Q,R\}\right\},$$

since

$$Q=P+2,R=P+3,R=Q+1.$$

Interpretation. Vertices represent entire weekly shift schedules; an (undirected) edge between two patterns indicates they differ by a uniform rotation of all daily blocks. Here $\{P,Q,R\}$ form a triangle, reflecting that any pattern can be reached from any other by some two-day rotation.

Theorem 2.4. 

For any group G and integer $n\ge 1$, the graph $P^{\left(n\right)}\left(G\right)$ is a 2-uniform n-SuperHyperGraph on G. Concretely, if one sets

$$V={\mathcal{P}}^n\left(G\right),E=E_n,$$

then $\left(V,E\right)$ satisfies the definition of an n-SuperHyperGraph, and its underlying simple graph is exactly $P^{\left(n\right)}\left(G\right)$.

Proof. 

By construction, $V={\mathcal{P}}^n\left(G\right)\subseteq {\mathcal{P}}^n\left(G_0\right)$ with $G_0=G$, and $E\subseteq \mathcal{P}\left(V\right)$. Hence $(V,E)$ is an n-SuperHyperGraph. Moreover, since every hyperedge in E has size 2 and corresponds precisely to the exponentiation relation ${\phi }_m^{\left(n\right)}$, the simple graph obtained by replacing each $\{A,B\}\in E$ by an undirected edge is exactly $P^{\left(n\right)}\left(G\right)$. □

Theorem 2.5 (Functoriality).

If $\psi :G\to H$ is a group isomorphism, then for each $n\ge 1$ there is a graph isomorphism

$${\Psi }^{\left(n\right)}:P^{\left(n\right)}\left(G\right)\stackrel{\cong }{\to }{P}^{\left(n\right)}\left(H\right),A\mapsto \psi \left(A\right)=\{\psi \left(a\right)\mid a\in A\}.$$

Proof. 

Since $\psi$ is bijective, so is ${\Psi }^{\left(n\right)}:{\mathcal{P}}^n\left(G\right)\to {\mathcal{P}}^n\left(H\right)$. Moreover, if $\{A,B\}$ is an edge in $P^{\left(n\right)}\left(G\right)$, then $B={\phi }_m^{\left(n\right)}\left(A\right)$ or vice versa for some m. Applying ${\Psi }^{\left(n\right)}$ yields

$$\psi \left(B\right)=\{\psi \left(b\right)\mid b\in B\}=\{\psi \left(a^m\right)\mid a\in A\}=\{\psi {\left(a\right)}^m\mid a\in A\}={\phi }_m^{\left(n\right)}\left(\psi \left(A\right)\right),$$

showing $\left\{\psi \right(A),\psi (B\left)\right\}$ is an edge of $P^{\left(n\right)}\left(H\right)$. Thus ${\Psi }^{\left(n\right)}$ preserves adjacency and is a graph isomorphism. □

Theorem 2.6 (Vertex-Transitivity).

For any finite group G and $n\ge 1$, the graph $P^{\left(n\right)}\left(G\right)$ is vertex-transitive under the action of G by conjugation on each level of the iterated powerset.

Proof. 

Let $g\in G$. Conjugation by g induces a permutation

$$c_g^{\left(k\right)}:{\mathcal{P}}^k\left(G\right)\to {\mathcal{P}}^k\left(G\right),A\mapsto gA{g}^{-1}=\{ga{g}^{-1}\mid a\in A\},$$

for each $k=1,2,\dots ,n$. These maps are bijections and satisfy

$$c_g^{(k+1)}\left({\phi }_m^{(k+1)}\left(A\right)\right)=\{c_g^{\left(k\right)}\left(a^m\right)\mid a\in A\}=\left\{{\left(c_g^{\left(k\right)}\left(a\right)\right)}^m\mid a\in A\right\}={\phi }_m^{(k+1)}\left(c_g^{(k+1)}\left(A\right)\right).$$

Hence conjugation commutes with iterated exponentiation, and so $\{A,B\}\in E_n$ if and only if $\{gA{g}^{-1},gB{g}^{-1}\}\in E_n$. This shows the induced permutation on $V_n={\mathcal{P}}^n\left(G\right)$ is an automorphism of $P^{\left(n\right)}\left(G\right)$. Since G acts transitively on itself by conjugation (and hence on its iterated powersets), $P^{\left(n\right)}\left(G\right)$ is vertex-transitive. □

Theorem 2.7 (Diameter Bound).

Let G be a finite group of exponent e (so $g^e=1$ for all $g\in G$). Then for any $n\ge 1$, the diameter of $P^{\left(n\right)}\left(G\right)$ is at most 2.

Proof. 

Given any two vertices $A,B\in V_n={\mathcal{P}}^n\left(G\right)$, if $\{A,B\}\in E_n$ we are done. Otherwise, consider the identity supervertex $\left\{1\right\}\in \mathcal{P}\left(G\right)\subset {\mathcal{P}}^n\left(G\right)$. Because each element of G has order dividing e,

$${\phi }_e^{\left(n\right)}\left(A\right)=\{a^e\mid a\in A\}=\left\{1\right\},{\phi }_e^{\left(n\right)}\left(B\right)=\left\{1\right\}.$$

Thus $\{A,\left\{1\right\}\}\in E_n$ and $\{B,\left\{1\right\}\}\in E_n$, so there is a path A–$\left\{1\right\}$–B of length 2. Hence $\mathrm{diam}\left(P^{\left(n\right)}\left(G\right)\right)\le 2$. □

2.2. Directed n-th Power Graphs

Directed n-th power graphs: vertices are n-fold powersets, with $A\to B$ if B equals the mth power of every element in A.

Definition 2.8 (Directed n-th Power Graph).

Under the same hypotheses, the directed n-th power graph of G is the digraph

$${\overrightarrow{P}}^{\left(n\right)}\left(G\right)=(V_n,A_n),V_n=P^n\left(G\right),$$

where

$$A_n=\left\{(A,B)\in V_n\times V_n|A\ne B,B=\{x^m\mid x\in A\}\mathrm{for}\mathrm{some}m\in \mathbb{N}\right\}.$$

Example 2.9 (Rotating Shift Schedules as a Directed 2-nd Power Graph).

We model a rotating shift schedule over a seven-day cycle using the directed 2-nd power graph.

Base group and first-level sets. Let

$$G={\mathbb{Z}}_7=\{0,1,2,3,4,5,6\},$$

with addition modulo 7 representing days of the week (e.g. $0=$Mon,$1=$Tue,…). A daily shift block is any nonempty subset $A\subseteq G$. For example,

$$A_1=\{0,1,2\}\left(\mathrm{Mon}\text{–}\mathrm{Wed}\mathrm{shift}\right),A_2=\{3,4\}\left(\mathrm{Thu}\text{–}\mathrm{Fri}\mathrm{shift}\right).$$

Second-level: weekly roster patterns. Elements of $P^2\left(G\right)=\mathcal{P}\left(\mathcal{P}\left(G\right)\right)$ are weekly patterns, i.e. collections of daily blocks. For instance,

$$P=\{A_1,A_2\}\mathrm{and}Q=\{A_1+2,A_2+2\},$$

where $A+2=\{x+2(mod7)\mid x\in A\}$ denotes shifting every block forward by two days:

$$A_1+2=\{2,3,4\},A_2+2=\{5,6\}.$$

Directed 2-nd power graph. Define

$$V_2=P^2\left(G\right),A_2=\left\{(P,Q)\in V_2\times V_2\mid Q=\{A+m\mid A\in P\}\mathrm{for}\mathrm{some}m\in \mathbb{N}\right\}.$$

Then ${\overrightarrow{P}}^{\left(2\right)}\left(G\right)=(V_2,A_2)$ is the directed 2-nd power graph. In particular, we have an arc $(P,Q)$ because Q is obtained by applying the same shift $m=2$ days to each daily block in P.

Interpretation. Vertices of ${\overrightarrow{P}}^{\left(2\right)}\left(G\right)$ represent entire weekly shift patterns, and a directed edge $P\to Q$ indicates that Q arises from P by uniformly advancing every daily block by m days. This captures the real-world operation of rotating shift schedules in a hierarchical (daily→weekly) framework.

Theorem 2.10. 

For any group G and integer $n\ge 1$:

1.

$P^{\left(1\right)}\left(G\right)$ coincides with the classical power graph $P\left(G\right)$ of Definition 1.2.

2.

${\overrightarrow{P}}^{\left(1\right)}\left(G\right)$ coincides with the directed power graph $\overrightarrow{P}\left(G\right)$ of Definition 1.4.

3.

The vertex sets satisfy $V_1\subseteq V_2\subseteq \dots \subseteq V_n\subseteq \dots$, so ${\left\{P^{\left(n\right)}\left(G\right)\right\}}_{n\ge 1}$ is a nested family reflecting the iterated powerset $P^n\left(G\right)$.

Proof.

1.

By the Definition, $P^1\left(G\right)=G$. Hence the adjacency rule in $P^{\left(1\right)}\left(G\right)$ reduces exactly to “$x^m=y$ or $y^m=x$” for $x,y\in G$, which is the condition defining $P\left(G\right)$.

2.

Likewise, ${\overrightarrow{P}}^{\left(1\right)}\left(G\right)$ has vertex set G and an arc $(x,y)$ precisely when $y=x^m$ for some m, matching Definition 1.4.

3.

From $P^k\left(G\right)=\mathcal{P}\left(P^{k-1}\left(G\right)\right)$, every element of $P^{k-1}\left(G\right)$ is also a (singleton) element of $P^k\left(G\right)$. Thus $V_{k-1}\subseteq V_k$ for all k, and the sequence of graphs $P^{\left(n\right)}\left(G\right)$ grows strictly (or stabilizes) with n. This nesting mirrors the iterated powerset structure.

□

Theorem 2.11. 

For any group G and $n\ge 1$, the digraph ${\overrightarrow{P}}^{\left(n\right)}\left(G\right)$ is a directed 2-uniform n-SuperHyperGraph. In particular, its arc-set $A_n$ defines the directed superedges of an n-SuperHyperGraph structure on $V_n={\mathcal{P}}^n\left(G\right)$.

Proof. 

Since $A_n\subseteq V_n\times V_n$ and each $(A,B)$ satisfies $B={\phi }_m^{\left(n\right)}\left(A\right)$, it endows $(V_n,A_n)$ with the structure of a directed hypergraph whose hyperedges all have size two and carry a direction. By Definition (specialized to directed arcs), this coincides with a directed 2-uniform n-SuperHyperGraph. □

Theorem 2.12 (Transitivity).

For any group G and $n\ge 1$, the directed n-th power graph ${\overrightarrow{P}}^{\left(n\right)}\left(G\right)$ is transitive: if $(A,B)$ and $(B,C)$ are arcs, then $(A,C)$ is also an arc.

Proof. 

Suppose $(A,B)\in A_n$ and $(B,C)\in A_n$. By definition there exist $m_1,m_2\in \mathbb{N}$ such that

$$B={\phi }_{m_1}^{\left(n\right)}\left(A\right),C={\phi }_{m_2}^{\left(n\right)}\left(B\right).$$

But then

$$C={\phi }_{m_2}^{\left(n\right)}\left({\phi }_{m_1}^{\left(n\right)}\left(A\right)\right)={\phi }_{m_1{m}_2}^{\left(n\right)}\left(A\right),$$

so $(A,C)\in A_n$. Hence the adjacency relation is transitive. □

Theorem 2.13 (Reachability to the Identity Supervertex).

Let G be a finite group of exponent e (i.e. $g^e=1$ for all $g\in G$). In ${\overrightarrow{P}}^{\left(n\right)}\left(G\right)$, every vertex $A\in P^n\left(G\right)$ has a directed path of length one to the singleton supervertex $\left\{1\right\}$.

Proof. 

For any $A\in V_n={\mathcal{P}}^n\left(G\right)$, consider

$$B={\phi }_e^{\left(n\right)}\left(A\right)=\{a^e\mid a\in A\}=\left\{1\right\}.$$

Since $A\ne B$ (unless $A=\left\{1\right\}$, in which case there is no self-loop), and $B={\phi }_e^{\left(n\right)}\left(A\right)$, the definition of arcs ensures $(A,\left\{1\right\})\in A_n$. Thus every supervertex reaches $\left\{1\right\}$ by a single directed edge. □

3. Conclusion and Future Work

In this paper, we have shown that the Power Graph of a group can be realized as a hypergraph and that the Directed Power Graph is a directed hypergraph. Furthermore, we introduced the n^th Power Graph and the Directed n^th Power Graph, and demonstrated that they form subclasses of SuperHyperGraphs and Directed SuperHyperGraphs, respectively. In future work, we plan to extend the concepts developed here to various frameworks of uncertainty, including Fuzzy Sets [36,37], Intuitionistic Fuzzy Sets [38], Hyperfuzzy Sets [39,40,41,42], SuperHyperFuzzy Sets [43,44], Neutrosophic Sets [45,46], and Plithogenic Sets [47,48,49].