On graphs related to conjugacy classes of groups
Given a finite group \(G\), the graphs \(\Gamma\) and \(\Gamma^*\) are defined as follows. The graph \(\Gamma\) has the noncentral conjugacy classes in \(G\) as vertices, two classes being adjacent if their sizes are not coprime. The graph \(\Gamma^*\) has the prime divisors of the sizes of conjugacy classes in \(G\) as vertices, where two such divisors are adjacent if they both divide the size of some conjugacy class in \(G\). It is proved that if \(\Gamma^*\) is connected then its diameter is at most 3 (a generalization of a similar result for solvable groups by S. Dolfi). Some bounds on the diameter of \(\Gamma\) are also proved.
scientific article; zbMATH DE number 590939