Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by \(\mathcal {E}(G)\) , is the sum of all absolute values of the eigenvalues of the adjacency matrix A(G). Recently, the concept of energy of a graph is extended to a self-loop graph. Let S be a subset of V(G) and \(\bar{S}=V(G)\backslash S\) . The graph \(G_{S}\) is obtained from the graph G by attaching a self-loop at each of the vertices of G which are in the set S. The energy of the self-loop graph \(G_{S}\) , denoted by \(\mathcal {E}(G_{S})\) , is the sum of all absolute eigenvalues of the adjacency matrix of \(G_{S}\) . In this paper, we first prove that if S is a vertex independent set of G and has no isolated vertices of G, then either \(\mathcal {E}(G_{S})>\mathcal {E}(G)\) or \(\mathcal {E}(G_{\bar{S}})>\mathcal {E}(G)\) . As a result, we confirm a conjecture on the energy of graphs with self-loops. Next, we establish a relation between \(\mathcal {E}(G_{S})\) and \(\mathcal {E}(G)\) , and we also obtain an upper bound for \(\mathcal {E}(G_{S})\) in terms of maximum degree. Furthermore, we derive an upper bound for the spread of energies of graphs \(G_{S}\) with \(\alpha \) self-loops and present an upper bound of Nordhaus–Gaddum type for energy of \(G_{S}\) . Finally, we construct pairs of equienergetic self-loop graphs of order 24n for all \(n\ge 1\) .
