Some New Results on Energy of Graphs with Self Loops

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 $E(G_{\sigma })={\displaystyle \sum _{i=1}^n\mid {\lambda }_i-{\displaystyle \frac{\sigma }{n}}\mid }$. It has been proved that if $\sigma =0orn$ 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.