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Let G be a finite group. An element g ∈ G is called a vanishing element of G if there exists an irreducible complex character χ of G such that χ(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. [ABSTRACT FROM PUBLISHER]

Let G be a finite group, and Irr( G) the set of irreducible complex characters of G. We say that an element g ∈ G is a vanishing element of G if there exists χ in Irr( G) such that χ( g) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G, and we define the prime graph on it, which we denote by Γ( G). Focusing on the class of solvable groups, we prove that Γ( G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ( G) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ( G). [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . This paper is a contribution to the study of the degree graph of G , that is, the prime graph built on the set cd ( G ) . Namely, we characterize finite groups whose degree graph has three independent vertices (i.e., three vertices that are pairwise non-adjacent). Our result turns out to be a generalization of several previously-known theorems concerning the structure of the degree graph. [ABSTRACT FROM AUTHOR]

Abstract: Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall -subgroups of G (which do exist) are metabelian. [Copyright &y& Elsevier]

Abstract: Given a finite group , denote by the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of , and for which two vertices of are adjacent if and only if they are not coprime numbers. In this note we prove that is a -regular graph if and only if it is a complete graph with three vertices, and is a -regular graph if and only if it is a complete graph with four vertices. [Copyright &y& Elsevier]

Abstract: Let G be a finite group, and let denote the prime graph built on the set of conjugacy class sizes of G. In this paper, we consider the situation when has “few complete vertices”, and our aim is to investigate the influence of this property on the group structure of G. More precisely, assuming that there exists at most one vertex of that is adjacent to all the other vertices, we show that G is solvable with Fitting height at most 3 (the bound being the best possible). Moreover, if has no complete vertices, then G is a semidirect product of two abelian groups having coprime orders. Finally, we completely characterize the case when is a regular graph. [Copyright &y& Elsevier]

Given a finite group G, denote by the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G, and set two vertices of to be adjacent if and only if they are not coprime numbers. In this note we prove that, if is a k-regular graph with k ≥ 1, then is a complete graph with vertices. [ABSTRACT FROM AUTHOR]