Abstract
Let \( G_{\sigma} \) be the graph obtained from a simple graph \( G \) of order \( n \) by adding \( \sigma \) self-loops, one self-loop at each vertex in \( S \subseteq V(G) \). Let \( \lambda_{1}(G_{\sigma}),\lambda_{2}(G_{\sigma}),\dots, \) \( \lambda_{n}(G_{\sigma}) \) be the eigenvalues of \( G_{\sigma} \). The energy of \( G_{\sigma} \), denoted by \( \mathscr{E}(G_{\sigma}) \), is defined as \( \mathscr{E}(G_{\sigma})=\sum\limits_{i=1}^{n}\left|\lambda_{i}(G_{\sigma})-\frac{\sigma}{n}\right| \). In this paper, using various analytic inequalities and previously established results, we derive several new lower and upper bounds on \( \mathscr{E}(G_{\sigma}) \).
