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 $x'\in x^G$ and $y'\in y^G$ 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 $D_{2n}$ ($n \geq 3$); 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. $D_{2n}$, $T_{4n}$, $U_{6n}$, $U_{(n, m)}$, $SD_{8n}$ and $V_{8n}$ such that their CCC-graphs are common neighborhood (signless) Laplacian hyperenergetic/borderenergetic.

Figures (8)

• Figure 1: CN-energies of $\Gamma_{D_{2n}}$, $n$ is odd
• Figure 2: CN-energies of $\Gamma_{D_{2n}}$, $n$ is even
• Figure 3: CN-energies of $\Gamma_{T_{4n}}$
• Figure 4: CN-energies of $\Gamma_{U_{(4,m)}}$, $m$ is odd
• Figure 5: CN-energies of $\Gamma_{SD_{8n}}$, $n$ is even

Theorems & Definitions (35)

• Theorem 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• proof
• Corollary 2.6