On the Energy of Graphs with Self-loops

Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by 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 S¯=V(G)\S. The graph GS 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 GS, denoted by E(GS), is the sum of all absolute eigenvalues of the adjacency matrix of GS. 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 E(GS)>E(G) or E(GS¯)>E(G). As a result, we confirm a conjecture on the energy of graphs with self-loops. Next, we establish a relation between E(GS) and E(G), and we also obtain an upper bound for E(GS) in terms of maximum degree. Furthermore, we derive an upper bound for the spread of energies of graphs GS with α self-loops and present an upper bound of Nordhaus–Gaddum type for energy of GS. Finally, we construct pairs of equienergetic self-loop graphs of order 24n for all n≥1.