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}.

Keywords

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