Groups with maximum vertex degree commuting graphs

Abstract

Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph $\hat{I}“(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that $\hat{I}“(G)$ is strong $k$ star free if the $k$ star graph is not a subgraph of $\hat{I}“(G)$. 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.

11 pages, 4 figures. To appear in Indian Journal of Pure and Applied Mathematics