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Common neighborhood Laplacian and signless Laplacian spectra and energies of commuting graphs

Firdous Ee Jannat, Rajat Kanti Nath

Abstract

In this paper, we compute common neighbourhood Laplacian spectrum, common neighbourhood signless Laplacian spectrum and their respective energies of commuting graph of some finite non-abelian groups including some AC-groups, groups whose central quotients are isomorphic to $Sz(2)$, $\mathbb{Z}_p\times \mathbb{Z}_p$ or $D_{2m}$. Our findings lead us to conclude that these graphs are CNL (CNSL)-integral. Additionally, we characterize the aforementioned groups such that their commuting graphs are CNL (CNSL)-hyperenergetic.

Common neighborhood Laplacian and signless Laplacian spectra and energies of commuting graphs

Abstract

In this paper, we compute common neighbourhood Laplacian spectrum, common neighbourhood signless Laplacian spectrum and their respective energies of commuting graph of some finite non-abelian groups including some AC-groups, groups whose central quotients are isomorphic to , or . Our findings lead us to conclude that these graphs are CNL (CNSL)-integral. Additionally, we characterize the aforementioned groups such that their commuting graphs are CNL (CNSL)-hyperenergetic.
Paper Structure (5 sections, 33 theorems, 172 equations)

This paper contains 5 sections, 33 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. CNL (CNSL)-spectrum and CNL (CNSL)-energy
  3. Some families of AC-group
  4. Groups whose central quotient is isomorphic to $Sz(2)$, $D_{2m}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$
  5. CNL (CNSL)-integral and CNL (CNSL)-hyperenergetic graphs

Key Result

Theorem 2.1

FR-2021 Let $\mathcal{G} = l_1 K_{m_1} \cup l_2 K_{m_2}\cup l_3 K_{m_3}$, where $l_iK_{m_i}$ denotes the disjoint union of $l_i$ copies of the complete graphs $K_{m_i}$ on ${m_i}$ vertices for $i = 1, 2, 3$. Then $\mathop{\mathrm{CNL-spec}}\nolimits(\mathcal{G})=\left\lbrace 0^{l_1 + l_2 + l_3}, (m_

Theorems & Definitions (61)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 51 more