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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]