Vertex energy distributions in regular graph structures

TL;DR

This paper investigates the vertex energy distribution in well-known regular graphs, providing explicit calculations for several classical graphs to understand their spectral properties.

Contribution

It introduces a method to compute vertex energy in regular graphs and applies it to several famous graph structures, expanding spectral graph theory knowledge.

Findings

Explicit vertex energy values for the Frucht, Desargues, Tutte-Coxeter, Heawood, Shrikhande, and Petersen graphs.

Demonstrates the application of vertex energy concepts to well-known regular graphs.

Provides insights into the spectral characteristics of these graphs.

Abstract

The energy of a vertex $v_i$ in a graph $G$ is defined as $\mathcal{E}_G(v_i) = |A|_{ii}$, where $A$ is the adjacency matrix of $G$, $A^*$ denotes the conjugate transpose of $A$, and $|A| = (AA^*)^{1/2}$. The total energy of the graph, $\mathcal{E}(G)$, is then the sum of the energies of all vertices: $\mathcal{E}(G) = \mathcal{E}_G(v_1) + \mathcal{E}_G(v_2) + \dots + \mathcal{E}_G(v_n)$. In this paper, we compute the vertex energy for several well-known regular graphs, including the Frucht graph, Desargues graph, Tutte-Coxeter graph, Heawood graph, Shrikhande graph, and Petersen graph.