On the vanishing prime graph of finite groups

Let G be a finite group. An element g ∈ G is called a vanishing element of G if there exists an irreducible complex character of G such that x(g) = 0. In this paper we study the vanishing prime graph Γ(G), whose vertices are the prime numbers dividing the orders of some vanishing element of G, and two distinct vertices p and q are adjacent if and only if G has a vanishing element of order divisible by pq. Among other things we prove that, similarly to what holds for the prime graph of G, the graph Γ(G) has at most six connected components. © 2010 London Mathematical Society.