Cycles and Bipartite Graph on Conjugacy Class of Groups

Abstract

Let G be a finite non abelian group and B(G) be the bipartite divisor graph of a finite group related to the conjugacy classes of G. We prove that B(G) is a cycle if and only if B(G) is a cycle of length 6 and G ≅ A × SL2(q), where A is abelian, and q ∈ {4,8}. We also prove that if G/Z(G) is simple, where Z(G) is the center of G, then B(G) has no cycle of length 4 if and only if G ≅ A × SL2(q), where q ∈ {4,8}.