Let G be a group and Z(G) be its center. We associate a commuting graph
Ξ(G), whose vertex set is GβZ(G) and two distinct vertices
are adjacent if they commute. We say that Ξ(G) is strong k star free
if the k star graph is not a subgraph of Ξ(G). In this paper, we
characterize all strong 5 star free commuting graphs. As a byproduct, we
classify all strong claw-free graphs. Also, we prove that the set of all
non-abelian groups whose commuting graph is strong k star free is finite.Comment: 12 pages, 4 figure