On bipartite divisor graphs for group conjugacy class sizes

In this paper we study various properties of a bipartite graph related to the sizes of the conjugacy classes of a finite group. It is proved that some invariants of the graph are rather strongly connected to the group structure. In particular we prove that the diameter is at most 6, and classify those groups for which the graphs have diameter 6. Moreover, if the graph is acyclic then the diameter is shown to be at most 5, and groups for which the graph is a path of length 5 are characterised.