Extreme Energy Of Quadtree With Diameter N-6

The energy of a simple graph is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix.And to determine the structure of graphs with extreme energy is one of the topics in the study of chemical graph theory.For a tree of a given diameter,finding a graph with extreme energy is an important part of the study of graph energy.There have been many studies on the extreme energy of tree with a given diameter.In the 1970's.the structure of the maximum and minimum energy of n-vertex trees is given by Gutman.In 2008,Li and Li given the structure of the third-largest energy tree.In 2011,Huo.Li.Shi determined the structure of the fourth-largest energy tree.In 2012,the structure of the maximal energy tree with diameter n-i-1 has been determined by Andriantiana(where i=1,2,3,4,6,8,10,12,14,16,18);In 2015,Suo.Ge.Huo gave the largest energy tree with diameter 4.This paper studies the maximal energy of trees with diameter n-6.First of all,the quadtree with diameter of n-6 is divided into two types of discussion.The structure of the tree with maximal energy is determined by using the method of the quasi-order in these two kinds of quadtrees respectively.It is found that the two maximal energy graphs of the two kinds of quadtrees are quasi-order incomparable.We solve this problem by using of Coulson integral formula,real analysis and combinatorial theory.Finally,the structure of the tree with the maximal energy of quadtrees with diameter n-6 is determined.However the energy of the trinary tree with diameter n-6 is completely sorted by using the method of the quasi-order.In this paper,a kind of special five-fork tree with diameter n-6 is tested and its maximal energy is found.However,it is found that there are a lot of quasi-order incomparable problem.In this paper,we make a conjecture about the structure of the maximal energy tree of a kind of special five-fork tree with the diameter n-6.it can be seen that finding a structure with a maximal energy in all the five-fork trees with diameter n-6 is a complicated question which needs to be resolved in the future.