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Common neighborhood energies and their relations with Zagreb index

Firdous Ee Jannat, Rajat Kanti Nath, Kinkar Chandra Das

Abstract

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph $\mathcal{G}$. We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that $\mathcal{G}$ is neither CNL-hyperenergetic nor CNSL-hyperenergetic if $\mathcal{G}$ is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph.

Common neighborhood energies and their relations with Zagreb index

Abstract

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph . We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that is neither CNL-hyperenergetic nor CNSL-hyperenergetic if is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph.
Paper Structure (5 sections, 26 theorems, 115 equations)

This paper contains 5 sections, 26 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Definition and connection with Zagreb index
  3. CN-(signless) Laplacian spectrum and CN-(signless) Laplacian energy
  4. Relation between various energies
  5. More bounds for $LE_{CN}(\mathcal{G})$ and $LE^+_{CN}(\mathcal{G})$

Key Result

Lemma 2.1

If $\mathcal{G} = \mathcal{G}_1 \sqcup \mathcal{G}_2 \sqcup \cdots \sqcup \mathcal{G}_k$ (that is, $\mathcal{G}_1, \mathcal{G}_2, \dots, \mathcal{G}_k$ are the disconnected components of $\mathcal{G}$), then and counting multiplicities.

Theorems & Definitions (46)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 36 more