Prime graphs on the set of vanishing class sizes

Abstract

Given a finite group G, an element is called a vanishing element of G if it is a zero of an irreducible character of G; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is termed as a vanishing class if that element is a vanishing element. In this paper, we prove that the prime graph on the set of vanishing class sizes of a given group has at most two connected components. Moreover, we examine the class of finite solvable groups whose prime graphs associated to the set of vanishing class sizes are disconnected.