Finite groups with star-free noncyclic graphs

TL;DR

This paper classifies finite noncyclic groups based on the structure of their noncyclic graphs, specifically those that do not contain a star graph $K_{1,n}$ for certain values of n.

Contribution

It provides a complete classification of finite noncyclic groups with noncyclic graphs free of star subgraphs $K_{1,n}$ for 3 ≤ n ≤ 6.

Findings

Identifies all such groups for each n in the specified range.

Characterizes the structure of noncyclic graphs that are $K_{1,n}$-free.

Provides a framework for understanding the relationship between group structure and graph properties.

Abstract

For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge between two distinct vertices $x$ and $y$ if $\langle x, y\rangle$ is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are $K_{1,n}$-free, where $3\leq n\leq 6$.