Some results on the non-commuting graph of a finite group

Let G be a metacyclic p-group, and let Z(G) be its center. The non-commuting graph Gamma(G) of a metacyclic p-group G is defined as the graph whose vertex set is G-Z(G), and two distinct vertices x and y are connected by an edge if and only if the commutator of x and y is not the identity. In this paper, we give some graph theoretical properties of the non-commuting graph Gamma(G) . Particularly, we investigate planarity, completeness, clique number and chromatic number of such graph. Also, we prove that if G(1) and G(2) are isoclinic metacyclic p-groups, then their associated graphs are isomorphic.