Graph energies and their applications

Abstract

The energy E(G) of a graph G is the sum of absolute values of the eigenvalues of the adjacency matrix of G. This spectral quantity was introduced in 1978 by Ivan Gutman, but its extensive research started only twenty-five years later. A large number (over a hundred) variants of graph energy have been proposed, based on matrices other than the adjacency matrix. Research of these graph energies is nowadays very active, resulting in well over a thousand publications. In recent years, more than two papers on graph energies appear each week. Graph energies found a remarkable number of various applications. In this paper, we outline some basic, mainly statistical, facts on the research of graph energies, and point out their main applications.