Cycles and bipartite graphs on conjugacy class of groups (Q991592)

Summary: 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 \(G\cong A\times\text{SL}_2(q)\), where \(A\) is Abelian, and \(q\in \{4,8\}\). We also prove that if \(G/Z(G)\) simple, where \(Z(G)\) is the center of \(G\), then \(B(G)\) has no cycle of length 4 if and only if \(G\cong A\times \text{SL}_2(q)\), where \(q\in \{4,8\}\).