On the intersection power graph of a finite group

Sudip Bera

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 $n\neq p^m$.


Keywords


automorphism group, intersection power graph, planar, p-groups

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DOI: http://dx.doi.org/10.5614/ejgta.2018.6.1.13

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