Some New Results on Energy of Graphs with Self Loops

TL;DR

This paper investigates the energy of graphs with self loops, providing new results and identifying a graph family where energy remains unchanged when adding self loops on a subset of vertices.

Contribution

It introduces a new graph family that maintains energy levels despite adding self loops on a subset of vertices, answering an open question.

Findings

Identified conditions where energy remains unchanged with self loops.

Constructed a graph family satisfying the energy invariance property.

Extended understanding of graph energy with self loops.

Abstract

The graph $G_\sigma$ is obtained from graph $G$ by attaching self loops on $\sigma$ vertices. The energy $ E(G_\sigma)$ of the graph $G_\sigma$ with order $n$ and eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$ is defined as $ E(G_\sigma)= \displaystyle \sum_{i=1}^n\left|\lambda_i-\dfrac{\sigma}{n}\right| $. It has been proved that if $\sigma=0\; or\; n$ then $ E(G)=E(G_\sigma) $. The obvious question arise: Are there any graph such that $E(G)=E(G_\sigma)$ and 0$<\sigma<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.