# On the Structure of Finite Groups Associated to Regular Non-Centralizer Graph

## TL;DR

This paper investigates the structure of finite groups whose non-centralizer or induced non-centralizer graphs are regular, revealing that such groups have quotients over their centers that are elementary 2-groups or p-groups.

## Contribution

It characterizes the structure of regular and induced regular finite groups based on their non-centralizer graph properties, a novel approach in group theory.

## Key findings

- Regular groups have quotients over their centers that are elementary 2-groups.
- Induced regular groups have quotients over their centers that are p-groups.
- The study provides structural insights linking graph regularity to group properties.

## Abstract

The non-centralizer graph of a finite group $G$ is the simple graph $\Upsilon_G$ whose vertices are the elements of $G$ with two vertices $x$ and $y$ are adjacent if their centralizers are distinct. The induced subgroup of $\Upsilon_G$ associated with the vertex set $G\setminus Z(G)$ is called the induced non-centralizer graph of $G$. The notions of non-centralizer and induced non-centralizer graphs were introduced by Tolue in \cite{to15}. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups as well as induced regular groups. Among the many obtained results, we prove that if a group $G$ is regular (resp. induced regular) then $G/Z(G)$ as an elementary $2-$group (resp. $p-$group).