Common neighbourhood spectrum and energy of commuting conjugacy class graph

TL;DR

This paper calculates the common neighbourhood spectrum and energy of commuting conjugacy class graphs for various finite non-abelian groups, revealing their spectral properties and energy characteristics.

Contribution

It introduces explicit computations of CN spectrum and energy for specific families of finite groups, expanding understanding of their spectral graph properties.

Findings

Groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ are CN-integral.

These graphs are not CN-hyperenergetic.

Results apply to groups with certain central quotients.

Abstract

In this paper we compute common neighbourhood (abbreviated as CN) spectrum and energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.