Groups with maximum vertex degree commuting graphs

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 \ 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 is an element of N is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4.