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Vertex energies of splitting and shadow graphs

Harshitha A, Madhumitha K, Sabitha D'Souza

Abstract

Graph energy has been widely investigated in spectral graph theory. However, the manner in which this energy is distributed among individual vertices has received attention only more recently, following the introduction of energy of a vertex by O. Arizmendi et al. in 2018. Since then, several studies have explored this concept. In the present work, we focus on splitting graphs and shadow graphs and study how the total energy of a graph is distributed among its vertices. We further establish explicit expressions for vertex energies in these derived structures and analyze their behavior in comparison with the underlying graph.

Vertex energies of splitting and shadow graphs

Abstract

Graph energy has been widely investigated in spectral graph theory. However, the manner in which this energy is distributed among individual vertices has received attention only more recently, following the introduction of energy of a vertex by O. Arizmendi et al. in 2018. Since then, several studies have explored this concept. In the present work, we focus on splitting graphs and shadow graphs and study how the total energy of a graph is distributed among its vertices. We further establish explicit expressions for vertex energies in these derived structures and analyze their behavior in comparison with the underlying graph.
Paper Structure (4 sections, 4 theorems, 29 equations)

This paper contains 4 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Vertex energy of $m-$splitting graph
  3. Vertex energy of the $m-$shadow graph
  4. Conclusion

Key Result

Theorem 1

Let $G$ be a graph with adjacency matrix $A$. Let $\mathrm{Spl}_m(G)$ be the $m$-splitting graph of $G$. Then the vertex energies satisfy

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Remark
  • Corollary
  • proof
  • Remark
  • ...and 6 more