On the regularity of a graph related to conjugacy classes of groups

A well-established research area in finite group theory consists in exploring the interplay between the structure of a group G and certain sets of positive integers, which are naturally associated to G. One of those sets, denoted by cs(G), is the set of conjugacy class sizes of G. In order to have a better understanding of the arithmetical structure of cs(G), it is useful to introduce two kind of graphs. One of them is the common divisor graph Γ(G), whose vertex set is cs(G) \ {1}, and two vertices are adjacent if and only if they are not coprime numbers. The other one is the prime graph ∆(G): in this case the vertices are the primes dividing some class size of G, and two vertices p, q are adjacent if there exists a class size of G that is divisible by pq. As stated in [1] we conjecture that , for every integer k ≥ 1, the graph Γ(G) is k-regular if and only if it is a complete graph with k + 1 vertices: in this poster we outline the fact that the conjecture is true for k ≤ 4. Every group considered in the following discussion will be a finite group.