On The Properties Of D_MAX-matrices Of Graphs

Radi（?）[36] introduced a novel distance matrix of graphs, D_MAX-matrix, and pointed that D_MAX-matrix might open new directions of exploration not only in Chemical Graph Theory but also possibly in Graph Theory on the role of dominant distances in graphs. The thesis mainly investigate the properties D_MAX-matrices of graphs and related topics.LetA（G） and D（G） be respectively the adjacency matrix and the distance matrix of a graph G. Then the D_MAX-matrix, which was constructed from the distance matrix D（G） of graphs so that in each row and each column it retains dominant distances, while other elements of the distance matrix are set to be zero. The D_MAXeigenvalues of G are those eigenvalues of D_MAX-matrix, and the D_MAX-spectrum of G is a multiset consisting of the D_MAX-eigenvalues together with their multiplicities.The largest eigenvalue of D_MAX-matrix of G is called D_MAX-spectrum radius. this thesis define the sum of the absolute value of all the eigenvalues of D_MAX-matrix called D_MAX-energy of graph.The thesis is organized as follows:1. In Chapter 1, we introduce the development of spectral graph theory and the research background of this thesis.2. In Chapter 2, we calculate the D_MAX-eigenvalues of graphs and study the D_MAX-matrix of a tree.3. In Chapter 3, we get out the upper and lower bounds of D_MAX-spectrum radius of graphs.4. In Chapter 4, we define the D_MAX-energy of a graph, and obtain the upper and lower bounds of D_MAX-energy.5. In Chapter 5, based on the properties of D_MAX-matrix, we research the relation between（r, d）-distance sum and diametrical graphs.