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: see text]-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: see text] and the determinant of the adjacency matrix [Formula: see text] The second bound relies on n, the minimum degree [Formula: see text] the independence number [Formula: see text] and [Formula: see text] 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: see text] and [Formula: see text] Next, we prove that for the general extended adjacency matrix [Formula: see text] with [Formula: see text] being the diagonal entries of [Formula: see text] the expression [Formula: see text] holds if and only if [Formula: see text] where [Formula: see text] Oboudi previously established that [Formula: see text] and proposed the problem of characterizing graphs with the spectrum [Formula: see text] for some non-negative integers [Formula: see text] with [Formula: see text] and [Formula: see text] 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: see text] We show that [Formula: see text] for all graphs having no eigenvalues in the interval (−1,1), where [Formula: see text] is the largest integer such that the star graph [Formula: see text] is an induced subgraph of G. We also prove that if the general extended matrix [Formula: see text] with [Formula: see text] for vertices [Formula: see text] 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.