The Energy Of The Tree With Diameter 5

The energy of a simple graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix,originally derived from the approximate estimate of the total?-electron energy of molecular orbitals in chemical graph.The greater the graph energy,the stronger the stability of the corresponding chemical molecules.To determine the structure of a graph with extreme energy is an important branch of graph energy research.In a tree with given diameter,finding the extremal energy graph is one of the research topics of graph energy,and there have been many results.This paper discusses the energy of the tree with diameter 5.The tree with diameter 5 has exactly two center vertices and these two vertices are adjacent.By quasi-order method,we find the maximal energy tree in each class of graphs that they respectively have same degree each center vertex,and same number of pendent vertices to each component of deleting edge between the two center vertices,respectively.This result is of great significance for eventually finding a maximal energy tree with a diameter of 5.For these kinds of trees,when the degree of the center vertices increase,their energies are quasi-order incompareable mutually.This quasi-order incomparable problem has not been solved yet.We get several such graphs' energies by computer software,and give a conjecture on these quasi-order incomparable problems.