On the intersection power graph of a finite group

Abstract

Given a group G, the intersection power graph of G, denoted by $\mathcal{G}_I(G)$, is the graph with vertex set G and two distinct vertices x and y are adjacent in $\mathcal{G}_I(G)$ if there exists a non-identity element $z\in G$ such that x^m=z=y^n, for some $m, n\in \mathbb{N}$, i.e. $x\sim y$ in $\mathcal{G}_I(G)$ if $\langle x\rangle\cap \langle y\rangle \neq \{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 $\mathcal{G}_I(G)$ is complete if and only if either G is cyclic p-group or G is a generalized quaternion group. Furthermore, $\mathcal{G}_I(G)$ is Eulerian if and only if |G| is odd. We characterize all abelian groups and also all non-abelian p-groups G, for which $\mathcal{G}_I(G)$ is dominatable. Beside, we determine the automorphism group of the graph $\mathcal{G}_I(\mathbb{Z}_n)$, when $n\neq p^m$.