Enhanced Power Graphs in Finite Group Theory

Updated 22 November 2025

Enhanced Power Graphs are graphs defined on group elements where two vertices are adjacent if the subgroup they generate is cyclic.
They interpolate between power graphs and commuting graphs, offering insights into subgroup structure, connectivity, and spectral invariants.
They facilitate group classification and reconstruction, leveraging decomposition theorems for finite nilpotent and abelian groups.

An enhanced power graph is a simple undirected graph associated with a group or semigroup, defined by cyclic-generation relations among the algebraic structure's elements. For a finite group $G$ , the enhanced power graph $\mathcal{G}_e(G)$ (also denoted as $\mathcal{P}_e(G)$ or $\mathcal{E}(G)$ ) encodes the adjacency structure where two distinct elements $x, y \in G$ are joined by an edge if and only if $\langle x, y\rangle$ is a cyclic subgroup. Enhanced power graphs interpolate between the power graph (adjacency via power relation) and the commuting graph (adjacency via commutativity), and their detailed paper connects group theory with structural graph theory, revealing precise correspondences between subgroup/element behavior and forbidden subgraph characterizations, automorphism structures, connectivity, and spectral graph invariants.

1. Definition and Core Properties

Let $G$ be a finite group. The enhanced power graph $\mathcal{G}_e(G)$ has:

Vertex set: $V = G$
Edge set: $\{x,y\} \in E$ if and only if $x \ne y$ and $\langle x, y\rangle$ is cyclic (i.e., $\exists z \in G$ such that $x, y \in \langle z\rangle$ ).

This definition naturally generalizes to infinite groups and to semigroups, where the adjacency is via inclusion in a monogenic or cyclic subsemigroup. The enhanced power graph always contains the power graph $\mathcal{G}(G)$ as a spanning subgraph and is always contained in the commuting graph $\mathcal{C}(G)$ : $\mathcal{G}(G) \subseteq \mathcal{G}_e(G) \subseteq \mathcal{C}(G)$ In the case of finite groups, the chain of inclusions is strict except under classical group-theoretic constraints (Aalipour et al., 2016, Surbhi et al., 14 Oct 2024).

The enhanced power graph is complete if and only if $G$ is cyclic (Bera et al., 2016). It is connected for all finite groups, and its diameter is always at most two for nontrivial $G$ (Panda et al., 2020).

2. Structural Decomposition and Isomorphism Theorems

For finite nilpotent groups, the enhanced power graph admits a canonical decomposition in terms of the Sylow $p$ -subgroups. The main decomposition theorem states: $\mathcal{G}_e(G) \cong \boxtimes_p \mathcal{G}_e(G_p)$ where $G_p$ are the Sylow $p$ -subgroups and $\boxtimes$ is the strong graph product (Zahirović et al., 2018, Mirzargar et al., 18 Mar 2025, Parveen et al., 2022). For two finite nilpotent groups $G, H$ , an isomorphism of their enhanced power graphs occurs if and only if their respective $p$ -Sylow enhanced power graphs are isomorphic for every $p$ .

This splitting property underpins classification results—specifically, the precise characterization of when the enhanced power graph is a complete group invariant within the nilpotent class. For example, for $G = Q_8 \times \mathbb{Z}_n$ ( $n$ odd), the enhanced power graph uniquely determines $G$ among all finite groups (Mirzargar et al., 18 Mar 2025).

For finite abelian $p$ -groups, enhanced power graphs are completely captured by the structure of a $p$ -semitree, and the entire graph is a strong product of these semitree factors (Zahirović et al., 2018).

3. Forbidden Subgraph Characterization and Graph-Theoretic Classes

Enhanced power graphs realize several well-studied graph-theoretic structures depending on the group:

Split/Threshold Graphs: $\mathcal{G}_e(G)$ is split or threshold if and only if $G$ is cyclic, dihedral, or elementary abelian 2-group (Ma et al., 2021).
Cographs/Chordal Graphs: For finite nilpotent groups, $\mathcal{G}_e(G)$ is a cograph/chordal iff $G$ has at most one non-cyclic Sylow subgroup. The only minimal forbidden subgraphs are $P_4$ and $C_4$ in this class (Ma et al., 2021, Bubboloni et al., 20 Oct 2025).
Diamond-Free (Block) Graphs: $\mathcal{G}_e(G)$ is diamond-free (hence a block graph) exactly if every nontrivial cyclic subgroup lies in a unique maximal cyclic subgroup (Bubboloni et al., 20 Oct 2025).
Quasi-threshold Graphs: For enhanced power graphs, quasi-threshold is equivalent to the cograph property (Bubboloni et al., 20 Oct 2025).

Simplicial vertices (vertices whose closed neighborhoods are cliques) correspond exactly to elements in unique maximal cyclic subgroups, and their structure is critical in cotree and elimination arguments for cograph and chordal status (Bubboloni et al., 20 Oct 2025).

4. Invariants and Extremal Properties

A variety of classical graph invariants are available in closed form:

Minimum degree: $\delta(\mathcal{G}_e(G))$ is the minimal $|M|-1$ over all maximal cyclic subgroups $M$ of $G$ (Panda et al., 2020, Parveen et al., 2022).
Connectivity: For non-cyclic abelian $p$ -groups, the vertex-connectivity is $1$; for non-cyclic abelian groups with at least two distinct Sylow primes, $\kappa \ge 2$ (Bera et al., 2020). For nilpotent groups with a unique non-cyclic Sylow, the minimal vertex-cut is precisely the product of the remaining cyclic Sylows (Parveen et al., 2022).
Independence number: $\alpha(\mathcal{G}_e(G))$ equals the number of maximal cyclic subgroups (Panda et al., 2020).
Matching, vertex/edge cover: Exact relations available for several classes through matching and Gallai identities (Panda et al., 2020).
Strong regularity: The proper enhanced power graph is strongly regular if and only if $G$ is cyclic or all maximal cyclic subgroups have the same order $m$ and meet trivially outside the identity (Parveen et al., 2022).
Rainbow connection number: Depends on the independence cyclic number ($\icn(G)$) and the arrangement (awning) of intersections among cyclic subgroups (Dupont et al., 2017).
$L(2,1)$ -labeling $\lambda$ -number: For nontrivial non-cyclic simple groups of order $n$ , $\lambda(\mathcal{G}_e(G)) = n$ . For nilpotent $G$ , $\lambda$ is tied closely to the number of dominating vertices plus $|G|$ (Parveen et al., 2022).

Graph parameters such as the Wiener index attain sharp bounds precisely when $G$ is cyclic (lower bound) or all cyclic subgroups have minimal order (upper bound); see (Parveen et al., 2022).

Spectra and eigenvalues for the distance and Laplacian matrices are available for classes such as dihedral, dicyclic, elementary abelian, and semidihedral groups, revealing multipartition and product structure (Arora et al., 2023, Parveen et al., 2021).

5. Enhanced Power Graphs and Group Reconstruction

Inverse and recognition problems have been explored:

Graph Isomorphism Invariance: For finite groups, the isomorphism types of directed power, (undirected) power, and enhanced power graphs coincide (Zahirović et al., 2018, Bošnjak et al., 2020).
Group Determination: For many small finite nilpotent groups, the enhanced power graph suffices to recover the group uniquely. However, non-isomorphic groups can share the same enhanced power graph, especially when their non-cyclic Sylows are of higher rank (see the non-abelian/abelian order $27$ example) (Mirzargar et al., 18 Mar 2025).
Reconstruction Algorithm: There is an explicit algorithm to reconstruct the enhanced power graph solely from the power graph, using a closed-twin counting function $f$ reflecting cyclic subgroup structure (Pote et al., 29 Oct 2025). Monotonicity of $f$ on the poset of cyclic subgroups underpins the combinatorial recovery (see table below).

Algorithm Step	Description
Power graph $X$ given	Compute all closed-twin functions $f(a)$
Non-adjacent pairs	Add edge $\{a,b\}$ in enhanced graph if $f$ and common neighbor criteria satisfied
Complete/Universal	Special handling for cyclic/complete graphs

This algorithm is canonical and operates group-independently in the finite case.

6. Extensions, Generalizations, and Open Problems

Several directions expand the theory and pose ongoing questions:

Automorphism-Orbit Enhanced Graphs: One may generalize the enhanced power graph by identifying elements via orbits under a subgroup $A \leq \operatorname{Aut}(G)$ , obtaining a quotient graph whose connectivity and diameter closely shadow the classical case, but for which completeness and emptiness criteria shift (Mohammadian et al., 21 Feb 2025).
Semigroup Theory: The enhanced power graph extends to semigroups, with characterizations for completeness, bipartiteness, regularity, tree/null structure, and planarity sharp in this wider setting (Dalal et al., 2021).
Geometric and Spectral Features: Metric dimension, resolving polynomials, and detour eccentricities can be computed for enhanced power graphs of notable non-abelian families, often reflecting the cyclic clustering of group elements (Parveen et al., 2021).
Extremal and Forbidden Substructures: Exact values for maximal neighborhood size in $p$ -groups and their characterizations in terms of group structure are known, with sharp lower bounds on neighborhood size and associated group-theoretic classifications (Lewis et al., 29 Aug 2024).
Open Problems: Full classification of finite groups with perfect enhanced power graphs remains open, especially for large alternating and simple groups and beyond two-prime nilpotency (Bošnjak et al., 2020). There are unresolved questions on the precise impact of automorphism group choices in the orbit-graph setting, the extent of the cograph property in simple groups, and the universality of the cograph property inheritance from the power graph (Bubboloni et al., 20 Oct 2025, Mohammadian et al., 21 Feb 2025).

7. Classification of Group Classes by Enhanced Power Graph Properties

A variety of group-theoretic properties correspond precisely to graph-theoretic signatures in the enhanced power graph, summarized:

Graph Property	Exact Group-Theoretic Criterion	Key References
Complete	$G$ cyclic	(Bera et al., 2016)
Eulerian	$\|G\|$ odd	(Bera et al., 2016)
Split/threshold	$G$ cyclic, dihedral, or elementary abelian 2-group	(Ma et al., 2021)
Cograph / Chordal	At most one non-cyclic Sylow subgroup	(Ma et al., 2021, Bubboloni et al., 20 Oct 2025)
Diamond-free/block	Each nontrivial cyclic subgroup lies in a unique maximal cyclic	(Bubboloni et al., 20 Oct 2025)
Perfect	At most two non-cyclic Sylow subgroups (nilpotent case)	(Zahirović et al., 2018, Bošnjak et al., 2020)
Cone property	Non-identity cone exists iff a cyclic Sylow or quaternion group	(Bera et al., 2016)
Planarity	All element orders $\le 4$	(Bera et al., 2016, Dalal et al., 2021)
Connectivity $=1$	Non-cyclic abelian $p$ -group	(Bera et al., 2020)

These exact correspondences enable immediate read-off of group algebraic properties from enhanced graph structure, and vice versa.

Through precise encoding of cyclic-generation relations, the enhanced power graph serves as a graph-theoretic invariant with rich algebraic and combinatorial content. Its systematic paper sharpens the distinction between various classes of groups, pinpoints graph invariants with algebraic meaning, and provides a setting where the interface of group and graph theory is especially transparent and tractable (Mirzargar et al., 18 Mar 2025, Zahirović et al., 2018, Ma et al., 2021, Bubboloni et al., 20 Oct 2025, Bera et al., 2016).