On 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 $A(G)$. It is the first eigenvalue-based topological molecular index and is related to the molecular orbital energy levels of π-electrons in conjugated hydrocarbons. Recently, the concept of energy of a graph is extended to a self-loop graph. Let S be a subset of $V(G)$. 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 matrix A(GS). Two non-isomorphic self-loop graphs are equienergetic if their energies are equal. Akbari et al. (2023)conjectured that there exist a subset S of $V(G)$ such that (GS) > E(G). In this paper, we confirm this conjecture. Also, we construct pairs of equienergetic self-loop graphs of order 24n for all n ≥ 1.