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.