Researches On The Adjacency Spectrum And Distance Spectrum Of Graphs

The theory of graph spectra is an active and important area in graph theory. There are extensive applications in the fields of quantum chemistry, statistical mechanics, computer science, communication networks and infor-mation science. In the theory of graph spectra, there are various matrices that are naturally associated with a graph, such as the adjacency matrix, the Laplacian matrix, the incidence matrix the distance matrix and the signless Laplacian matrix and so on. One of the main problems of graph spectra theory is to determine precisely how, or whether properties of graphs are reflected in the algebraic properties of the above matrices.(i) In Chapter1, we first look back the evolvement of graph theory. Then, we introduce some definitions and notations for the corresponding questions. Lastly, we introduce the backgrounds and research progresses of some questions on graph spectra theory which we study in this paper.(ii) In Chapter2, we first give the upper of the spectral radius with given order and chromatic number (arc connectivity). We also characterize the di-graphs which have the maximal spectral radius in the set of graphs with given order and chromatic number (arc connectivity). In the last, we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs. Furthermore, we show that any strongly connected bicyclic digraph is determined by the spectrum.(iii) In Chapter3, we first give upper and lower bounds on q1(G)-μ1(G), q1(G)-λ1(G) and μ1(G)-λ1(G), moreover, the extremal graphs which attains the upper and lower bounds are characterized. In addition, we give a sharp lower bound of q1(G)+q2(G). Secondly, we give some sharp upper and lower bounds on the largest and least eigenvalues of graphs when vertices are removed. We also prove some conjectures in [2,3] involving the spectral radius, diameter and matching number. Furthermore, we characterize the extremal graph which attains the minimum least eigenvalue among all quasi-tree graphs. Lastly, we present sharp upper bounds of λ2for connected graphs and connected bipartite graphs in this paper. Moreover, the extremal graphs are completely characterized.(iii) In Chapter4, we first characterize the extremal digraphs with min-imum distance spectral radius among all digraphs with given vertex connec-tivity (arc connectivity) and the extremal graphs with minimum distance spectral radius among all graphs with given edge connectivity. In the last, we characterize all connected graphs with λn(D)=-2. Furthermore, we characterize all connected graphs of diameter2with exactly three different D-eigenvalues when λ1(D) is not an integer. We also conjecture that the complete k-partite graph is determined by its D-spectrum.