On bipartite divisor graphs for group conjugacy class sizes. (Q1020951)

Many authors have considered the influence of arithmetic structure, such as conjugacy class sizes or character degrees, on the algebraic structure of a finite group. Graphs have been introduced to illustrate this arithmetic structure. Let \(X\) be a set of positive integers, then the `prime vertex graph' \(\Delta(X)\) of \(X\) is defined as follows. The vertex set of \(\Delta(X)\) is given by the set of prime divisors of elements of \(X\), and two vertices \(p\) and \(q\) are joined if there exists \(x\in X\) divisible by \(pq\). Alternatively, the `common divisor graph' \(\Gamma(X)\) of \(X\), has vertex set \(X\setminus 1\) (\(X\) need not contain the element 1) and two vertices are joined if they have a common divisor greater than 1. The relationship between these two graphs has been recently elucidated by the introduction of a third graph \(B(X)\), the `bipartite divisor graph', in a paper by \textit{M. A. Iranmanesh} and \textit{C. E. Praeger}, [Bipartite divisor graphs for integer subsets, (submitted)]. The vertex set of \(B(X)\) is the disjoint union of the vertex sets of \(\Delta(X)\) and \(\Gamma(X)\) and the edge set is given by \(\{{p,n}:p\in V(\Delta(X))\), \(n\in V(\Gamma(X))\), and \(p\mid n\}\). In the paper under review the authors consider \(B(X)\) where \(X\) is the set of conjugacy class sizes of a finite group \(G\), they then denote the graph by \(B(G)\). They prove that \(B(G)\) has diameter at most 6 and they classify those groups for which the graphs have diameter 6. They also consider the case when the graph is acyclic. In this case the graph has diameter at most 5, and groups for which the graph is a path of length 5 are characterised. The proofs apply many known results alongside careful analysis of particular cases.