The intersection graph of a group

Abstract

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

On the intersection graph of a finite group

“…For finite groups, Akbari and his coauthors in [1] proved the following lemmas: Lemma 2.1. Let G be a finite group.…”

“…Xuanlong Ma in [9] gave an upper bound for the diameter of intersection graphs of finite simple groups. Akbari et al in [1] classified finite groups whose intersection graphs are triangle-free. Also, they determined all finite groups with null intersection graphs or complete intersection graphs.…”

Equal‐Square Graphs Associated with Finite Groups

“…A large amount of literature is devoted to study the graphs associated to finite groups, for instance, commuting graphs [9][10][11][12][13][14][15], noncommuting graphs [16,17], intersection graphs [18], prime graphs [19][20][21], conjugacy class graphs [22], power graphs [23][24][25], inverse graphs [26], quadratic residues graphs [27], order divisor graphs [28], and square graphs [29].…”

The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

“…The intersection graph of is an undirected simple (without loops and multiple edges) graph whose vertex-set consists of all nontrivial proper subgroups of for which two distinct vertices H and K of are adjacent if ⋂ is a nontrivial subgroup of . This kind of graph has been studied by researchers; we refer the reader to see [2][3][4][5][6]. Let be any graph.…”