On the Conjecture Related to the Energy of Graphs with Self–Loops

Abstract

Let Gσ be a graph obtained by attaching a self-loop, or just a loop, for short, at each of σ 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(Gσ) of a corresponding graph Gσ, for 1 ≤ σ ≤ 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 Gσ, respectively.