The Energy Of Trees With Given Diameter

The energy of graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix. The definition of graph energy comes from total π-electron energy of H¨uckel molecular orbital approximation. Graph energy is an important branch of chemical graph theory. There is a close relationship between the graph energy and molecular graph chemical properties. The energy of a graph can reflect the thermodynamic stability of its corresponding conjugated compounds chemical structure. When the energy of a graph is larger, the thermodynamic stability of the corresponding compound is stronger. In history,the energy of graph has been widely studied due to the widely used in chemical. In particular, the energy of tree whose structure is relatively simple received extensive attention of researchers in recent years.A lot of achievements have been made about extremal energy trees with given diameter. Among all trees with n vertices, Gutman proved that the path Pnand the star K1,n-1are the maximal and minimal energy trees, respectively. Obviously, they also respectively have the maximal and minimal diameters. In 2005, Yan and Ye further determined the structures of minimal energy trees among all n vertices trees with a given diameter. In 2012, Andriantiana determined the maximal energy trees with diameter n- i- 1, where i = 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18. In addition, Ou determined the structures of the maximal energy trees with center degrees of t and diameter 4. And he also determined the structures the maximal energy trees of diameter 5 with a perfect matching. For other conditions, there is no result.On the basis of predecessors’ work, this paper continues to push forward this research. The main contents include:First of all, all energies of trees with diameter 3 are ordered by using the method of quasi-order. In 2008, Ou classified trees with diameter 4 according to the degrees of center vertex, and determined the structures of maximal energy trees in every kind. Finding the maximal energy tree among all trees with diameter 4 is a quasi-order-incomparable problem, and this problem is unsolved. In the paper, the structures of maximal energy trees with diameter 4 are completely determined by using quasi-order and the Taylor expansion approximation of the real analysis.And then, the conditions of diameter 5 and n- 6 are discussed. Firstly, some structure characteristics of maximal energy trees with diameter 5 are determined through two graph operations. And through observation and calculation for small order number trees, a guess is given about maximal energy trees with diameter5. Then, in the case of diameter of n- 6, the maximal energy trees in one kind of them are determined by using the method of quasi-order.