# The energy and spectrum of non commuting graph

## TL;DR

This paper investigates the spectral properties and energy of non-commuting graphs associated with dihedral groups and their products, providing explicit calculations and extending known results in algebraic graph theory.

## Contribution

It computes the energy, Laplacian energy, and spectrum of non-commuting graphs for dihedral groups and their products, offering new explicit formulas and extending previous work.

## Key findings

- Calculated the energy and spectrum of non-commuting graphs of D2n.
- Extended results to non-commuting graphs of D2n × D2n and G × H.
- Provided explicit formulas for energies of these graphs.

## Abstract

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph {\Gamma}(G) of G is a graph with vertex set is non central elements of G and two vertices x, y are adjacent if and only if they are commute. In this paper we calculate the energy, Laplacian energy and spectrum of non-commuting graph of dihedral group D2n. Also we will obtain the energy of non-commuting graph of D2n \times D2n and G \times H, where G is a non-abelian finite group and H is an abelian finite group