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 $E(G_\sigma)= \displaystyle \sum_{i=1}^n\left|\lambda_i-\dfrac{\sigma}{n}\right|$. 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.