# On the Non-Commuting Graph of Dihedral Group

## TL;DR

This paper studies various graph-theoretic properties of the non-commuting graph of dihedral groups, including indices, eccentricities, and mean distances, enhancing understanding of their structural characteristics.

## Contribution

It introduces new calculations of indices and distances for the non-commuting graph of dihedral groups, expanding the algebraic graph theory literature.

## Key findings

- Computed detour index, eccentric connectivity, and total eccentricity polynomials.
- Determined the mean distance of the non-commuting graph for dihedral groups.
- Provided explicit formulas for these graph invariants.

## Abstract

For a nonabelian group G, the non-commuting graph $\Gamma_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_G$ are adjacent if they do not commute in $G$. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of non-commuting graph on $D_{2n}$. We also find the mean distance of non-commuting graph on $D_{2n}$.