Common neighborhood (signless) Laplacian spectrum and energy of CCC-graph

Abstract

In this paper, we consider commuting conjugacy class graph (abbreviated as CCC-graph) of a finite group $G$ which is a graph with vertex set $\{x^G:x\in G\setminus Z(G)\}$ (where $x^G$ denotes the conjugacy class containing $x$) and two distinct vertices $x^G$ and $y^G$ are joined by an edge if there exist some elements $x^{\prime }\in x^G$ and $y^{\prime }\in y^G$ such that they commute. We compute common neighborhood (signless) Laplacian spectrum and energy of CCC-graph of finite non-abelian groups whose central quotient is isomorphic to either ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$ (where $p$ is any prime) or the dihedral group $D_{2n}$ ($n\ge 3$); and determine whether CCC-graphs of these groups are common neighborhood (signless) Laplacian hyperenergetic/borderenergetic. As a consequence, we characterize certain finite non-abelian groups viz. $D_{2n}$, $T_{4n}$, $U_{6n}$, $U_{(n,m)}$, $S{D}_{8n}$ and $V_{8n}$ such that their CCC-graphs are common neighborhood (signless) Laplacian hyperenergetic/borderenergetic.

31 pages