A Study on Graph Energy

Abstract

ABSTRACT newlineSpectral graph theory is the study of properties of a graph, related to the newlinecharacteristic polynomial, the eigenvalues and the eigen vectors of the matrices newlineassociated to the graph. One of the most remarkable chemical applications of newlinespectral graph theory is based on the close correspondence between the eigenval- newlineues of the graph and the molecular orbital energy levels of and#960;-electrons in conju- newlinegated hydrocarbons. newlineIf G is the molecular graph of a conjugated hydrocarbon with n vertices and newlineand#955;1, and#955;2, . . . , and#955;n are its eigenvalues, then in the Huckel molecular orbital (HMO) newlinetheory, the energy of the ith molecular orbital is given by Ei = and#945; + and#955;iand#946; where and#945; newlineand and#946; are constants. For and#945; = 0 and and#946; = 1, the and#960;-electron orbital energies and newlinethe eigenvalues of the graph are equal. newlineConsequently the total and#960;-electron energy of a molecule is the sum of energies newlineof all the and#960;-electrons which are in that respective molecule. i.e., E = newlinePn newlinei=1 newlinehiEi = newlinePn newlinei=1 newlinehiand#955;i where hi is the number of electrons in the ith molecular orbital with energy newlineEi. By Pauli s Exclusion Principle, hi = 2, 1, or 0. In the most of chemically newlinerelevant cases, hi = 2 whenever and#955;i gt 0 and hi = 0 whenever and#955;i lt 0, imply that newlineE = 2 newlineP newline+ newlineand#955;i where newlineP newline+ newlineindicates the summation over the positive eigenvalues. newlineThe fact that the sum of all eigenvalues of a graph is zero, enables to arrive newlineat the definition of graph energy, introduced by Ivan Gutman in 1978 as E(G) = newlinePn newlinei=1 |and#955;i| where and#955;1, and#955;2, . . . , and#955;n are the eigenvalues of the adjacency matrix of the newlinegraph G. A large number of bounds for the energy of a graph in terms of the newlinenumber of the vertices and edges have been reported in the literature. Also newlinegraphs with maximal and minimal energies are conjectured. newlineIn the past decade, interest in the notion of graph energy has increased and newlinemany different versions have been conceived. In 2006, Gutman and Zhou defined newlinethe Laplacian energy of a graph as the sum of the absolute deviations from the newlinemean degree of eigenvalues of the Laplacian matrix. Similar variants of graph newlineenergy a