Intersection Power Graph in Finite Group Theory

• The intersection power graph is defined on a finite group where non-identity vertices are adjacent if their generated cyclic subgroups intersect nontrivially, highlighting shared nontrivial powers.
• It refines the power graph concept by incorporating common power relations and is used to compare standard and enhanced power graphs through structures like the difference graph.
• The analysis provides explicit classifications, connectivity bounds, and clique properties that offer insights into both abelian and non-abelian group behaviors.

In finite-group theory, the intersection power graph of a finite group $G$ is the graph on vertex set $G$ in which two distinct non-identity vertices $x,y$ are adjacent exactly when their cyclic subgroups meet nontrivially; by convention, the identity is adjacent to all other vertices. This graph sits naturally between subgroup-intersection and power-relation viewpoints: for $x,y\neq e$, adjacency is equivalent to the existence of $k,\ell\ge 1$ with $x^k=y^\ell\neq e$, whereas in the ordinary power graph adjacency requires one vertex to be a power of the other. Consequently the power graph is a spanning subgraph of the intersection power graph, and recent work isolates their discrepancy through the difference graph $D(G)$, obtained from $E(I(G))\setminus E(P(G))$ and then deleting isolated vertices (Bera et al., 4 Sep 2025).

1. Definitions and immediate consequences

Let $G$ be a finite group with identity $e$. The three graphs considered on the common vertex set $G$0 are as follows (Bera et al., 4 Sep 2025).

Graph	Vertex set	Adjacency
Power graph $G$1	$G$2	$G$3 iff $G$4 for some $G$5 or $G$6 for some $G$7
Intersection power graph $G$8	$G$9	For distinct $x,y$0, $x,y$1 iff $x,y$2; by convention, $x,y$3 is adjacent to all vertices
Difference graph $x,y$4	$x,y$5, then isolated vertices deleted	$x,y$6

For non-identity vertices, the condition $x,y$7 is equivalent to the existence of $x,y$8 such that $x,y$9. This makes $x,y\neq e$0 sensitive to common nontrivial powers rather than to direct power containment. Every power-graph edge is therefore an intersection-power-graph edge, so $x,y\neq e$1 is always a spanning subgraph of $x,y\neq e$2.

The identity behaves uniformly in $x,y\neq e$3 and $x,y\neq e$4. Since $x,y\neq e$5 for any $x,y\neq e$6, one has $x,y\neq e$7. By the stated convention, also $x,y\neq e$8. In the difference graph, however, every edge incident with $x,y\neq e$9 occurs in both $k,\ell\ge 1$0 and $k,\ell\ge 1$1, so $k,\ell\ge 1$2 has degree $k,\ell\ge 1$3; it is therefore removed in $k,\ell\ge 1$4, or remains isolated in the undeleted version.

A basic conceptual point is that $k,\ell\ge 1$5 records precisely the pairs $k,\ell\ge 1$6 for which cyclic subgroups intersect nontrivially but neither element is a power of the other. This suggests that the study of intersection power graphs is often most informative when carried out relative to the power graph, rather than in isolation.

2. Equality with the power graph and null-difference phenomena

The most rigid situation is $k,\ell\ge 1$7, equivalently the difference graph is empty. One reduction theorem states: if either $k,\ell\ge 1$8 cannot be generated by two elements or $k,\ell\ge 1$9, then $x^k=y^\ell\neq e$0 is null if and only if $x^k=y^\ell\neq e$1 is null for every proper subgroup $x^k=y^\ell\neq e$2 (Bera et al., 4 Sep 2025). The proof uses the fact that adjacency in $x^k=y^\ell\neq e$3 for a pair $x^k=y^\ell\neq e$4 depends only on $x^k=y^\ell\neq e$5, together with the observation that when $x^k=y^\ell\neq e$6 and $x^k=y^\ell\neq e$7, any element of $x^k=y^\ell\neq e$8 centralizes both $x^k=y^\ell\neq e$9 and $D(G)$0, hence lies in the center and must be trivial.

Prime-order elements are especially restrictive. If $D(G)$1 is prime, then $D(G)$2 is isolated in the undeleted difference graph. In particular, if every element has prime-power order, that is, if $D(G)$3 is an EPPO group, then $D(G)$4 is null. A further consequence is that if $D(G)$5 is null, then $D(G)$6 is a cograph; the converse need not hold in general.

Among finite simple groups, the null-difference condition is rare. The only non-abelian finite simple groups with null difference graph are precisely those $D(G)$7 for which, with $D(G)$8, each of $D(G)$9 and $E(I(G))\setminus E(P(G))$0 is either a prime power or the product of two primes (Bera et al., 4 Sep 2025). This places the equality $E(I(G))\setminus E(P(G))$1 under explicit number-theoretic control in that family.

The comparison with other power-graph variants is instructive. The equality $E(I(G))\setminus E(P(G))$2 enhanced power graph is known to hold precisely for EPPO groups, whereas equality $E(I(G))\setminus E(P(G))$3 is rarer and is governed by the null-difference classification in the finite-group setting. A plausible implication is that intersection-power adjacency is substantially more permissive than enhanced-power adjacency once nontrivial subgroup intersections occur without a direct power containment relation.

3. Isolated vertices, connectedness, and diameter

A complete description of isolated vertices is given under the hypothesis $E(I(G))\setminus E(P(G))$4 (Bera et al., 4 Sep 2025). Writing $E(I(G))\setminus E(P(G))$5 for the set of generators of $E(I(G))\setminus E(P(G))$6, the isolated vertices in the deleted $E(I(G))\setminus E(P(G))$7 are:

• If $E(I(G))\setminus E(P(G))$8 with $E(I(G))\setminus E(P(G))$9 primes, then $G$0.
• If $G$1 with $G$2, then $G$3.
• Otherwise, $G$4.

Even without the centrality hypothesis, every non-identity element of prime order is isolated in the undeleted difference graph. In cyclic groups, generators are also isolated except in the case $G$5.

The main connectivity theorem states that if $G$6 is finite with $G$7, then $G$8 is connected if and only if $G$9. Moreover, whenever connected, $e$0. In the cyclic case with $e$1, $e$2 is disconnected if and only if $e$3, and whenever connected its diameter is again at most $e$4.

Sharper bounds are available for broad non-cyclic families. If $e$5 is neither cyclic nor a product $e$6 with $e$7, then $e$8 is connected and $e$9. The proof constructs large cliques from central elements of distinct prime orders and shows that every vertex is within distance at most $G$00 of such a clique. If $G$01 with $G$02 and $G$03, then $G$04 is connected with $G$05 (Bera et al., 4 Sep 2025).

These results distinguish the deleted difference graph from both $G$06 and $G$07. The intersection power graph itself contains the identity as a universal vertex by convention, whereas the deleted difference graph removes this trivial source of connectivity and still often remains connected with uniformly bounded diameter.

4. Cyclic groups, $G$08-groups, and explicit examples

For a cyclic group $G$09, adjacency becomes an order-theoretic condition. Since $G$10 is cyclic of order $G$11, one has

$G$12

By contrast,

$G$13

Hence in the undeleted difference graph there is an edge between $G$14 and $G$15 exactly when $G$16 but neither $G$17 nor $G$18 (Bera et al., 4 Sep 2025).

Three small cases illustrate the contrast. For $G$19, orders are $G$20, and every pair with $G$21 also satisfies a divisibility relation, so $G$22. For $G$23, both $G$24 and $G$25 are complete, so again $G$26. For $G$27, difference edges occur between elements of orders $G$28 and $G$29, since $G$30 but neither order divides the other; thus every element of order $G$31 is adjacent in $G$32 to every element of order $G$33, while pairs such as $G$34 are non-adjacent because $G$35.

Within cyclic groups, generators form an empty subgraph in $G$36. Moreover, if one element of a generator set $G$37 is adjacent to one element of $G$38, then the entire bipartite block between $G$39 and $G$40 is complete. This converts many clique and induced-subgraph questions into combinatorics on divisor families.

For finite $G$41-groups there is a precise classification of emptiness of the difference graph: $G$42 is empty if and only if either $G$43 is cyclic of order $G$44, or $G$45 is a union of at least three proper cyclic $G$46-subgroups. The latter condition is presented in the paper as

$G$47

The proof idea is that within each cyclic subgroup both $G$48 and $G$49 are complete, while across distinct cyclic subgroups the intersections are trivial, forcing $G$50 (Bera et al., 4 Sep 2025).

5. Non-abelian behavior and structural invariants

Non-abelian examples show that intersection power graphs detect phenomena invisible to power graphs. In $G$51, all non-identity elements have prime order $G$52 or $G$53, so every such element is isolated in $G$54; hence the deleted difference graph is empty. In $G$55, the two $G$56-cycles are adjacent because they generate the same order-$G$57 subgroup, but this edge already lies in $G$58, so no difference edge survives.

For dihedral groups, $G$59. The elements outside the cyclic rotation subgroup have order $G$60 and are isolated in $G$61, so all difference edges come from the cyclic part. For generalized quaternion groups $G$62 with $G$63, the contrast is sharper: $G$64 is complete, but $G$65 is not complete unless $G$66 is a cyclic $G$67-group. In $G$68, the unique involution is isolated in $G$69, and any element outside the index-$G$70 cyclic subgroup is adjacent in $G$71 to any element of that subgroup of order not equal to $G$72; thus $G$73 is non-empty and highly connected once the involution is removed (Bera et al., 4 Sep 2025).

Several global graph-theoretic properties of $G$74 are now known. For finite nilpotent groups with $G$75, $G$76 is perfect if and only if $G$77. For groups with $G$78, $G$79 is bipartite if and only if $G$80 with distinct primes $G$81 and $G$82. The special case $G$83 yields the empty graph, while for $G$84 the bipartition separates orders $G$85 from orders $G$86.

The difference graph is Eulerian for every finite group. In the undeleted graph every vertex has even degree because the neighborhood of a vertex $G$87 splits into disjoint generator classes $G$88, each contributing $G$89, and $G$90 is even for $G$91. This universal Eulerian property is one of the most robust invariants currently known (Bera et al., 4 Sep 2025).

6. Clique theory, universality, twin reduction, and terminological variants

If $G$92 is cyclic of order $G$93, any maximal clique is determined by a set $G$94 of divisors of $G$95, with vertex set

$G$96

The admissible divisor families differ across the three graph types. In $G$97, $G$98 is a clique iff $G$99 is a chain under divisibility. In $x,y$00, $x,y$01 is a clique iff $x,y$02 is an intersecting family, meaning pairwise $x,y$03. In $x,y$04, $x,y$05 is a clique iff $x,y$06 is an intersecting Sperner family, that is, no inclusion among members and pairwise $x,y$07. The paper indicates that computing $x,y$08 reduces to a weighted extremal problem over intersecting Sperner families with weights $x,y$09, invoking a weighted version of the de Bruijn–Tengbergen–Kruyswijk Sperner-type theorem on divisors (Bera et al., 4 Sep 2025).

The class of difference graphs is universal: every finite graph is an induced subgraph of $x,y$10 for some cyclic squarefree $x,y$11. The construction uses Sperner families and assigns distinct primes to ground elements. This implies that induced-subgraph complexity for difference graphs is, in a precise sense, unrestricted.

The same work also uses twin reduction computationally. Twin vertices are vertices with identical neighborhoods; false twins satisfy $x,y$12, while true twins satisfy $x,y$13. Twin reduction collapses each equivalence class to a single representative. It is used to simplify large difference graphs and to study structural properties, including examples such as $x,y$14. Although not axiomatized further in the paper, it is described as the standard graph-theoretic quotient by the relation of having the same neighborhood, and it preserves properties like connectedness while often aiding symmetry recognition (Bera et al., 4 Sep 2025).

Several open directions remain explicit. The paper asks for a connectivity classification when $x,y$15 and whether $x,y$16 can exceed $x,y$17; for a full classification of isolated vertices beyond the cyclic and $x,y$18 cases; for a characterization of perfect difference graphs beyond the nilpotent case with $x,y$19; and for a complete classification of finite groups with bipartite $x,y$20.

The term “intersection power graph” also appears in other mathematical literatures, but with different meanings. In random graph theory, inhomogeneous random intersection graphs have been studied as sparse models with power-law degrees and clustering: vertices are adjacent when they share an attribute in a bipartite affiliation graph, yielding explicit asymptotic formulas for degree distributions, degree-degree distributions, clustering, and assortativity (Bloznelis, 2014, Bloznelis et al., 2013). In a distinct topological-graph construction on a finite discrete space $x,y$21 with $x,y$22, the graph denoted $x,y$23 has vertex set $x,y$24 and edges joining disjoint subsets; the paper proves $x,y$25, $x,y$26, $x,y$27, $x,y$28, connectedness, and for $x,y$29, $x,y$30 and $x,y$31 (Omran et al., 2022). These usages are terminologically related but structurally separate from the finite-group intersection power graph.