The commuting conjugacy class graphs of finite groups with a given property

Abstract

Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ ( G ) {\Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ ( G ) {\Gamma(G)} are connected by an edge if there exist elements x ∈ X {x\in X} and y ∈ Y {y\in Y} such that x ⁢ y = y ⁢ x {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ ( G ) {\frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p 2 ⁢ q {p^{2}q} , is determined.