On lower bounds for the energy of graphs

The concept of the energy of a graph was introduced by Gutman in 1978, inspired by the Hückel molecular orbital theory, which approximates the total (Formula presented.) -electron energy of a conjugated hydrocarbon molecule using the energy of its molecular graph. Let G be a graph on n vertices with m edges. In this paper, we first present two lower bounds for the energy of a graph. The first bound is based on the order n, the maximum degree (Formula presented.) and the determinant of the adjacency matrix (Formula presented.) The second bound relies on n, the minimum degree (Formula presented.) the independence number (Formula presented.) and (Formula presented.) We also determine extremal graphs for our lower bounds. In addition, we obtain an upper bound for the graph energy in terms of size m, the maximum degree (Formula presented.) and (Formula presented.) Next, we prove that for the general extended adjacency matrix (Formula presented.) with (Formula presented.) being the diagonal entries of (Formula presented.) the expression (Formula presented.) holds if and only if (Formula presented.) where (Formula presented.) Oboudi previously established that (Formula presented.) and proposed the problem of characterizing graphs with the spectrum (Formula presented.) for some non-negative integers (Formula presented.) with (Formula presented.) and (Formula presented.) Here, we provide a generalization of Oboudi’s lower bound for the energy of graphs, and then characterize graphs with the above spectrum when (Formula presented.) We show that (Formula presented.) for all graphs having no eigenvalues in the interval (−1,1), where (Formula presented.) is the largest integer such that the star graph (Formula presented.) is an induced subgraph of G. We also prove that if the general extended matrix (Formula presented.) with (Formula presented.) for vertices (Formula presented.) has exactly one positive eigenvalue, then one of the components of G is a complete multipartite graph, while all other components, if any, are isolated vertices.