Some New Results on Energy of Graphs with Self Loops

Abstract

The graph $G_{\ddot{I}}ƒ$ is obtained from graph $G$ by attaching self loops on $\ddot{I}ƒ$ vertices. The energy $E(G_{\ddot{I}}ƒ)$ of the graph $G_{\ddot{I}}ƒ$ with order $n$ and eigenvalues $\hat{I}{»}_1,\hat{I}{»}_2,\dots ,\hat{I}{»}_n$ is defined as $E(G_{\ddot{I}}ƒ)=\sum _{i=1}^n\left|\hat{I}{»}_i-\frac{\ddot{I}}{ƒ}n\right|$. It has been proved that if $\ddot{I}ƒ=0orn$ then $E(G)=E(G_{\ddot{I}}ƒ)$. The obvious question arise: Are there any graph such that $E(G)=E(G_{\ddot{I}}ƒ)$ and 0$<\ddot{I}ƒ<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.

Published in Journal of Mathematical Chemistry (2023)