Abstract
Let \(G_{\sigma}\) be a graph obtained by attaching a self-loop, or just a loop, for short, at each of \(\sigma\) chosen vertices of a given graph \(G\). Gutman et al. have recently introduced the concept of the energy of graphs with self-loops, and conjectured
that the energy \(E(G)\) of a graph \(G\) of order \(n\) is always strictly less than the energy \(E(G_{\sigma})\) of a corresponding graph \(G_{\sigma}\), for \(1\leq\sigma\leq n-1\). In this paper, a simple set of graphs which disproves
this conjecture is exposed, together with some remarks regarding the standard deviations of the (adjacency) eigenvalues of \(G\) and \(G_{\sigma}\), respectively.