On the intersection power graph of a finite group

Abstract

Given a group G, the intersection power graph of G, denoted by G_I(G), is the graph with vertex set G and two distinct vertices x and y are adjacent in G_I(G) if there exists a non-identity element z ∈ G such that x^m=z=yⁿ, for some m, n ∈ N, i.e. x ∼ y in G_I(G) if ⟨x⟩ ∩ ⟨y⟩ ≠ {e} and e is adjacent to all other vertices, where e is the identity element of the group G. Here we show that the graph G_I(G) is complete if and only if either G is cyclic p-group or G is a generalized quaternion group. Furthermore, G_I(G) is Eulerian if and only if ∣G∣ is odd. We characterize all abelian groups and also all non-abelian p-groups G, for which G_I(G) is dominatable. Beside, we determine the automorphism group of the graph G_I(Z_n), when n ≠ p^m.