Vertex energies of splitting and shadow graphs

TL;DR

This paper investigates how the total energy of splitting and shadow graphs is distributed among vertices, providing explicit formulas and analyzing their behavior compared to the original graph.

Contribution

It introduces explicit expressions for vertex energies in splitting and shadow graphs and compares their distribution to that of the original graph.

Findings

Derived explicit formulas for vertex energies in splitting and shadow graphs.

Analyzed the distribution of vertex energies in these graphs.

Compared vertex energy behavior with underlying original 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.