Superpower Graph in Algebraic Graph Theory

• Superpower Graph is a graph-theoretic construction in group theory that generalizes ordinary power graphs by incorporating order-divisibility and shared cyclic subgroup intersections.
• It encompasses related variants such as the order supergraph, order superpower graph, and generalized power graph, each emphasizing different levels of algebraic proximity.
• These constructions enable precise combinatorial, spectral, and structural analyses, and serve as effective detectors for properties like cyclicity and p-group structures.

Superpower graph denotes several closely related but non-identical constructions, principally in algebraic graph theory, all of which enlarge the adjacency relation of the ordinary power graph of a group. In one standard usage, the superpower graph $\mathcal{S}_G$ is the simple undirected graph on the elements of a finite group $G$ in which distinct vertices $a,b$ are adjacent exactly when $o(a)\mid o(b)$ or $o(b)\mid o(a)$ (Mir et al., 9 Aug 2025). Closely related papers use the terms order supergraph $\mathcal{S}(G)$ and order superpower graph for the same order-divisibility construction (Ashrafi et al., 2017, Parveen et al., 2024). A different generalization is the generalized power graph $GP(G)$, whose vertices are the elements generating proper cyclic subgroups and whose edges record nontrivial intersections of those cyclic subgroups (Jafarzadeh et al., 2018). This suggests that “Superpower Graph” is best treated as an umbrella term for several supergraph-type extensions of the power graph rather than as a uniquely standardized object.

1. Core definitions and relations

The ordinary power graph, enhanced power graph, order supergraph, and generalized power graph all retain the same underlying intuition—graphically encoding algebraic proximity among group elements—but they do so at different levels of coarseness.

Construction	Vertex set	Adjacency
Power graph $P(G)$	$G$	one element is a power of the other
Enhanced power graph $PE(G)$	$G$0	the two vertices lie in a common cyclic subgroup
Order supergraph / order superpower graph $G$1	$G$2	$G$3 or $G$4
Generalized power graph $G$5	$G$6	$G$7

For the ordinary power graph, two distinct vertices $G$8 are adjacent if $G$9 or $a,b$0 for some positive integers $a,b$1. In the torsion-free setting, the directed version $a,b$2 orients this relation by placing an arc $a,b$3 whenever $a,b$4 for some nonzero integer $a,b$5 (Cameron et al., 2017). A commonly studied identity-deleted variant is the “new power graph” $a,b$6, the induced subgraph on the non-identity elements (Jafari, 2015).

The enhanced power graph $a,b$7 broadens adjacency from literal power containment to common membership in some cyclic subgroup. The order supergraph broadens it further by discarding subgroup containment entirely and retaining only divisibility of element orders. For finite $a,b$8, the ordinary power graph is a spanning subgraph of both $a,b$9 and $o(a)\mid o(b)$0 (Parveen et al., 2024). The generalized power graph $o(a)\mid o(b)$1 broadens adjacency in another direction: distinct vertices $o(a)\mid o(b)$2 are adjacent when

$o(a)\mid o(b)$3

The authors of the generalized-power-graph paper note that, “with a slight difference in vertices,” one can regard $o(a)\mid o(b)$4 as a subgraph of $o(a)\mid o(b)$5 (Jafarzadeh et al., 2018).

A further terminological point is important. In the order-divisibility literature, the graph

$o(a)\mid o(b)$6

is called the order supergraph in one paper (Ashrafi et al., 2017), the order superpower graph in another (Parveen et al., 2024), and the superpower graph $o(a)\mid o(b)$7 in a spectral study (Mir et al., 9 Aug 2025). The underlying definition is the same.

2. Order-divisibility superpower graphs of finite groups

For a finite group $o(a)\mid o(b)$8, the order supergraph $o(a)\mid o(b)$9 is always connected, because $o(b)\mid o(a)$0 divides every element order. In fact,

$o(b)\mid o(a)$1

This near-complete structure is one reason the graph is easier to analyze combinatorially than the ordinary power graph (Ashrafi et al., 2017).

Several exact group-theoretic characterizations are known. The equality

$o(b)\mid o(a)$2

holds if and only if $o(b)\mid o(a)$3 is cyclic. Completeness is even more rigid: $o(b)\mid o(a)$4 The graph is planar if and only if

$o(b)\mid o(a)$5

and the proper order supergraph $o(b)\mid o(a)$6, obtained by deleting the identity, is planar if and only if

$o(b)\mid o(a)$7

The same paper proves that $o(b)\mid o(a)$8 is perfect, and characterizes the bipartite and tree cases by

$o(b)\mid o(a)$9

with the same condition for $\mathcal{S}(G)$0 to be a tree (Ashrafi et al., 2017).

Minimal connectivity properties impose further arithmetic restrictions. For a finite group $\mathcal{S}(G)$1,

$\mathcal{S}(G)$2

If $\mathcal{S}(G)$3 is a finite group of full exponent, then

$\mathcal{S}(G)$4

For finite nilpotent groups,

$\mathcal{S}(G)$5

holds if and only if either $\mathcal{S}(G)$6 is a $\mathcal{S}(G)$7-group or $\mathcal{S}(G)$8, where $\mathcal{S}(G)$9 is a $GP(G)$0-group with $GP(G)$1 (Parveen et al., 2024). These theorems make the order-divisibility superpower graph a precise detector for $GP(G)$2-group structure.

The same order-divisibility graph has also been analyzed through explicit graph-representation questions. The order supergraph $GP(G)$3 is a line graph if and only if $GP(G)$4 is an EPPO-group and $GP(G)$5 is divisible by at most two distinct primes. The proper order supergraph $GP(G)$6 is a line graph if and only if $GP(G)$7 or $GP(G)$8 is an EPPO-group. For any $GP(G)$9,

$P(G)$0

In the dominatable case, $P(G)$1 is a line graph exactly when $P(G)$2 is a $P(G)$3-group, or $P(G)$4 is divisible by exactly two primes and every element order is square-free (Manisha et al., 2023).

3. The generalized power graph $P(G)$5

The generalized power graph $P(G)$6 is a distinct superpower-type construction. Its vertex set is

$P(G)$7

and distinct vertices $P(G)$8 are adjacent if

$P(G)$9

The graph therefore records shared cyclic structure rather than order divisibility or literal power inclusion (Jafarzadeh et al., 2018).

Completeness of $G$0 is highly restrictive. For a torsion abelian group $G$1, $G$2 is complete if and only if

$G$3

for some prime $G$4. If $G$5 is torsion-free abelian, then $G$6 is complete if and only if $G$7 is a subgroup of $G$8. In the finite non-abelian case,

$G$9

for some $PE(G)$0, where $PE(G)$1 is the generalized quaternion group. The infinite non-abelian theory is partial: a nilpotent torsion-free group with complete $PE(G)$2 must be abelian, and a locally finite torsion non-abelian group with complete $PE(G)$3 is a generalized quaternion group (Jafarzadeh et al., 2018).

For $PE(G)$4-groups, $PE(G)$5 has a particularly strong decomposition theorem. If $PE(G)$6 is a $PE(G)$7-group, then every connected component of $PE(G)$8 is complete, and the number of components is exactly the number of distinct subgroups of order $PE(G)$9 in $G$00. This yields a “clique decomposition” indexed by order-$G$01 subgroups (Jafarzadeh et al., 2018).

Planarity is even rarer than completeness. For finite abelian groups, $G$02 is planar if and only if $G$03 is one of:

1. an elementary abelian $G$04-group,
2. an elementary abelian $G$05-group,
3. an elementary abelian $G$06-group,
4. $G$07,
5. $G$08.

For non-abelian $G$09-groups with planar generalized power graph, the results are partial but sharp in low-prime cases. If $G$10, planarity forces exponent $G$11; if $G$12, a finite non-abelian $G$13-group with planar $G$14 must satisfy

$G$15

These classifications show that the generalized power graph is extremely sensitive to subgroup intersection geometry (Jafarzadeh et al., 2018).

4. Spectral and matrix-theoretic analysis

The order-divisibility superpower graph has also been studied by spectral methods. For a finite group $G$16, the superpower graph $G$17 in this literature is the simple undirected graph whose vertices are the elements of $G$18, with adjacency determined by divisibility of element orders: $G$19 The block structure induced by order classes produces explicit adjacency, Laplacian, and $G$20-spectral formulas in certain non-abelian families (Mir et al., 9 Aug 2025).

A representative case is $G$21, where $G$22 is prime. The possible element orders are

$G$23

and the vertex set decomposes into classes

$G$24

with sizes

$G$25

The adjacency spectrum satisfies

$G$26

together with three remaining eigenvalues given as the roots of a cubic equation. The Laplacian spectrum is

$G$27

The same paper studies Nikiforov’s interpolating matrix

$G$28

for both $G$29 and $G$30, again exploiting equitable block structure induced by element-order classes (Mir et al., 9 Aug 2025).

This spectral program is structurally consistent with earlier order-supergraph work. Since order supergraphs are controlled by the divisibility pattern of element orders, the natural quotient matrices are indexed by order classes rather than by subgroup chains. A plausible implication is that the most explicit spectra should occur precisely in groups where the order partition is small and highly regular.

5. Adjacent variants in the power-graph program

The modern theory of superpower graphs sits inside a broader power-graph program that includes directed, proper, and non-group variants. These are not identical to superpower graphs, but they provide the surrounding framework in which supergraph-type constructions arose.

For torsion-free groups, the central issue is often whether the undirected power graph determines the directed power graph. For finite groups, if $G$31, then $G$32. For infinite groups this can fail, but strong positive results are known for $G$33, for $G$34 up to global reversal of orientation, for $G$35, and for torsion-free nilpotent groups of class at most $G$36 (Cameron et al., 2017). These results clarify how much of the algebraic “power” relation survives in an undirected encoding.

The identity-deleted graph $G$37 was introduced to remove the trivial dominance of the identity vertex. For a finite group $G$38, the paper defines

$G$39

so that the vertices are the non-identity elements and adjacency remains “one is a power of the other.” For $G$40-groups,

$G$41

and for finite nilpotent groups, $G$42 is connected exactly when $G$43 is cyclic, generalized quaternion, or not a $G$44-group (Jafari, 2015). These results are closely parallel to generalized-power-graph connectivity theorems, but the objects are distinct.

Power graphs have also been extended beyond groups. In gyrogroups, the power graph $G$45 is defined by the same adjacency condition $G$46 or $G$47. For the gyrogroup $G$48 of order $G$49, the graph consists of a complete graph on the vertices of $G$50 together with $G$51 pendant vertices attached only to the identity. The graph is planar if and only if $G$52, and it is not Hamiltonian (Singh et al., 2022). This shows that the power-graph formalism is portable to non-associative algebraic settings.

6. Terminological diffusion outside algebra

Outside group theory, the phraseology of “power graph” and “superpower” appears in several unrelated literatures. These usages are conceptually distinct from algebraic superpower graphs and are best treated as homonyms or analogies rather than instances of the same object.

In graph visualization, a power graph is a compressed representation of a dense graph that groups nodes with similar neighborhoods into modules. A node-to-module edge means that the node is connected to every member of the module, and a module-to-module edge represents a complete bipartite connection between the two modules. This construction is explicitly described as a lossless compression method for clearer visualization of dense graphs, and the paper proves that computing the optimal power graph with only one module is NP-hard (Dwyer et al., 2013). This notion has no connection to powers of group elements.

In statistical methodology, PowerGraph refers to a machine-learning framework for graphing and predicting statistical power over a multivariate parameter space. The paper argues that the object of interest is not a one-dimensional power curve but a power manifold, and proposes a supervised neural-network approach called PNN, supplemented by PCA-based feature engineering and optional transfer learning (Mulay et al., 2021). Despite the name, this is not a graph-theoretic superpower graph.

In geometry processing, the relevant object is the superpower contour, not a superpower graph. The paper on SpUDD states explicitly that “the phrase ‘superpower graph’ does not appear in the paper.” The formal object is

$G$53

a subset of power-diagram faces used as a proxy for unsigned surface reconstruction (Wang et al., 21 Apr 2026).

Category theory provides yet another unrelated use of “superpower.” The paper on coalgebraic graphs introduces several superpower set constructions that allow arbitrarily nested subsets, making possible graphs with recursively nested nodes and edges incident to edges. The resulting coalgebra categories are shown to be $G$54-adhesive when the relevant functor preserves pullbacks along monomorphisms (Padberg, 2017). Here “superpower” modifies a powerset construction, not a graph based on power relations.

A final source of ambiguity is rhetorical rather than terminological. In representation learning on heterogeneous information networks, the HINormer paper says that Graph Transformers have a “superpower” because global-range attention allows larger-coverage message passing, even across the whole graph (Mao et al., 2023). This is a descriptive phrase, not the name of a graph object.

Taken together, these literatures show that “Superpower Graph” is standardized only locally. In finite-group theory it usually refers to an order-divisibility supergraph of the power graph, or to a nearby generalization such as the generalized power graph. In other fields, the same wording either denotes an unrelated construction or does not occur at all.