Abstract: A well-established area of research within finite group theory is the study of the connection between the structure of a group G and the arithmetical properties of certain sets of positive integers related to it. In general, a tool that turns out to be very useful for studying arithmetical features of a set X of positive integers is the prime graph ∆(X): this is the simple undirected graph having the prime divisors of the numbers in X as vertices, and the edges are pairs {p,q} such that the product pq divides some number in X.
In this talk, I will discuss some classical as well as some more recent results when X=cs(G), the set of conjugacy class sizes of a finite group G. Besides, I will mention some similarities with the prime graph built on the set X=cd(G) of degrees of irreducible characters of G.