Groups with maximum vertex degree commuting graphs

Abstract

Let G be a finite non-abelian group and Z(G) be its center. We associate a commuting graph \(\Gamma (G)\) to G, whose vertex set is \(G\setminus Z(G)\) and two distinct vertices are adjacent if they commute. In this paper we prove that the set of all non-abelian groups whose commuting graph has maximum vertex degree bounded above by a fixed \(k \in {\mathbb {N}}\) is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4.