Abstract
The energy of a graph \( G_S \) with \( n \) vertices and \( \sigma \) self-loops is defined as \( \varepsilon(G_S) = \sum_{i=1}^{n}|\lambda_i -\frac{\sigma}{n}| \), where the eigenvalues of the adjacency matrix of \( G_S \) are \( \lambda_1, \lambda_2,
\dots, \lambda_n \). In this article, we have established some upper and lower bounds for the energy of such a graph. Those new bounds involve parameters like number of vertices (n), number of edges (m), number of self-loops (\( \sigma)
\), maximum vertex degree (\( \Delta \)), and minimum vertex degree (\( \delta \)). We show that for \( 1\leq \sigma
< n \), the quantity \( |\lambda_i -\frac{\sigma}{n}| \) is always greater than \( 0 \), and using that fact we establish a lower bound. We have compared and concluded that the new bounds are either better than the existing bounds or incomparable to a
few bounds obtained by some researchers recently.
