Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups

TL;DR

This paper proves a conjecture stating that non-commuting graphs uniquely determine the order of non-abelian finite groups, confirming that isomorphic graphs imply groups have the same size.

Contribution

The paper provides a proof that the non-commuting graph of a non-abelian finite group uniquely determines its order, resolving a previously open conjecture.

Findings

Non-commuting graphs distinguish non-abelian finite groups by order.

Isomorphic non-commuting graphs imply groups have equal size.

The conjecture by Abdollahi-Akbari-Maimani is confirmed.

Abstract

The non--commuting graph $\Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(\Gamma(G))$ of $\Gamma(G)$ is $G\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. For non--abelian finite groups $G$ and $H$ it is conjectured that if $\Gamma(G) \cong \Gamma(H)$, then $|G|=|H|$. We prove the conjecture.