ScholarWorks@성균관대학교

초록

Let G be a simple graph on n vertices with vertex set V (G). The energy of G, denoted by epsilon(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<overline> = V (G)\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 GS, denoted by epsilon(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 epsilon(G(S)) > epsilon(G) or epsilon(G(S<overline>)) > epsilon(G). As a result, we confirm a conjecture on the energy of graphs with self-loops. Next, we establish a relation between epsilon(G(S)) and epsilon(G), and we also obtain an upper bound for epsilon(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 >= 1.

키워드