The Union Of Three Paths’ Extreme Energy And Energy Ordering Of A Class Graph

Let G be an undirected simple graph of order n and A(G) its adjacency matrix. The energy of graph G is de?ned as the sum of the absolute values of all eigenvalues of A(G). The concept was derived from the H ¨uckel molecular orbital(HMO) approximation for total π-electrons energy. Graph energy is an important branch of chemical graph theory. There exists a close relationship between graph energy and chemical properties of molecules. The greater the graph energy is,the stronger corresponding thermodynamic stability of the compound is.In 1940, Charles Coulson et al obtained Coulson integral formula when they calculated the energy of a kind of chemical molecule. The formula establishes a directed linkage between the energy of a graph and its characteristic polynomial. By utilizing earlier results, in 1978, Gutman abstracted the original chemical concept, and put forward the mathematical de?nition like the one of today. The de?nition is available not only to chemical molecule graphs, but also to general graphs. The Coulson integral formula plays an important role in the study of graph energy, and it was ?rst used to calculate the energy of simple graphs whose adjacency matrices are real symmetric, and the sum of eigenvalues is zero. After then, Gutman, Shao et al modi?ed the formula, and Shao generalized the de?nition of graph energy and the corresponding integral formula to the case of real or complex polynomials. By primary real analytic method and Coulson integral formula of real symmetric matrix, the second chapter of this paper obtains the integral formula for the energy of real matrix whose eigenvalues are real.The quasi-order relation in bipartite graphs was de?ned by Gutman in1977. By the method of quasi-order relation, many problems of extreme energies in bipartite graphs and certain other graph were e?ciently settled. All along, this method was an important approach to solving most problems of extreme energies and energy ordering. In 1986, by the quasi-order method, Gutman obtained the ordering of energy for n vertices graphs which are union of two paths. The result plays an important role in calculating graph energy, especially in the extreme energy of tree and bipartite graph. But there is not a general method for ordering the union of three paths’ energy. The third chapter gives the union of three paths’ extreme energy by quasi-order method and an ordering of energy for a special class of graphs. The extreme energies include the maximal, the second maximal, the third maximal energy and the minimal, the second minimal, the third minimal energy. The fourth chapter gives the ordering of energy for a class graph by quasi-order method.