On the Non-Commuting Graph of the Group $U_{6n}$

TL;DR

This paper investigates the properties of the non-commuting graph of the group U_{6n}, including its structural parameters and polynomial invariants, providing new formulas and insights into its combinatorial characteristics.

Contribution

It introduces explicit formulas for the resolving polynomial and other graph invariants of the non-commuting graph of U_{6n}, expanding understanding of its combinatorial structure.

Findings

Derived the general formula for the resolving polynomial.

Calculated the independent number, clique number, and chromatic number.

Determined the detour index, eccentric connectivity, and total eccentricity.

Abstract

A non-commuting graph of a finite group $G$ is a graph whose vertices are non-central elements of $G$ and two vertices are adjacent if they don't commute in $G$. In this paper, we study the non-commuting graph of the group $U_{6n}$ and explore some of its properties including the independent number, clique and chromatic numbers. Also, the general formula of the resolving polynomial of the non-commuting graph of the group $U_{6n}$ are provided. Furthermore, we find the detour index, eccentric connectivity, total eccentricity and independent polynomials of the graph.