The Intersection Centralizer Graph of Finite Groups

Let G be a group. Th e intersection centralizer graph of G, denoted by ΓC(G), is the graph whose vertex set is the set of all noncentral elements of G and two distinct vertices x and y are adjacent if and only if CG(x)∩CG(y) ≠ Z(G), where CG(x) is the centralizer of element x in G. In this paper we show that for AC-group G, ΓC(G) is disjoint union of |cent(G)|-1 complete blocks. Moreover we show that the intersection centralizer graph of two isoclinic groups with the same order of their centers, are isomorphic. And so if ΓC(G) be triangle-free graph, then G is a soluble group.