### 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=y^{n}, 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}.

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

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ISSN: 2338-2287

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