On Equienergetic Graphs and Graph Energy of Some Standard Graphs with Self loops

Let $G_S$ be the graph of order $n$ and containing $\sigma$ self-loops. The energy $E(G_S)$ of graph $G_S$ is defined as $E(G_S)=\displaystyle\sum_{i=1}^{n}\bigg\lvert\lambda_i-\dfrac{\sigma}{n}\bigg\rvert$, where $\lambda_1, \lambda_2, \dots, \lambda_n$ be the eigenvalues of the adjacency matrix of $G_S$. Two non-isomorphic graphs $G_1$ and $G_2$ of the same order are said to be equienergetic if they have same energy. The proposed research is an effort to expand the concept of equienergetic graphs from simple graphs to graphs having self-loops. In the present work, we have obtained a pair of equienergetic graphs and the energy of complete graphs as well as complete bipartite graphs with self loops.