Non-commuting graphs of nilpotent groups

TL;DR

This paper investigates the structure of non-commuting graphs of non-abelian nilpotent groups, proving that isomorphic irregular graphs imply equal group orders, revealing a link between graph properties and group order.

Contribution

It establishes that non-abelian nilpotent groups with irregular isomorphic non-commuting graphs must have the same order, a novel connection between graph isomorphism and group size.

Findings

Irregular isomorphic non-commuting graphs imply equal group orders.

Non-abelian nilpotent groups' structure is reflected in their non-commuting graphs.

Graph properties can determine algebraic group characteristics.

Abstract

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph $\Gamma_G$ associated to $G$ is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then $|G|=|H|$.