Groups with maximum vertex degree commuting graphs

TL;DR

This paper characterizes all groups whose commuting graphs do not contain a star with five leaves and classifies all strong claw-free graphs, revealing finiteness in certain non-abelian group classes.

Contribution

It provides a complete characterization of strong 5 star free commuting graphs and classifies all strong claw-free graphs, establishing finiteness results for non-abelian groups with these properties.

Findings

Characterization of all strong 5 star free commuting graphs

Classification of all strong claw-free graphs

Finiteness of non-abelian groups with strong k star free commuting graphs

Abstract

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