Some Results On The Spectral Radius Of Non-regular Graphs

The graph spectral theory is an active and important area in graph the-ory. 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 spectral, there are various matrices that are naturally associated with a graph, such as the adjacency matrix, the Signless Laplacian matrix, the incidence matrix, the distance matrix and so on. One of the main problems of graph spectral theory is to determine precisely how, or whether, properties of graphs are reflected in the algebraic properties (especially properties about eigenvalues, such as spectral radius, spectral uniqueness, spread, energy and so on) of such matrices.Among the above mentioned matrices of graphs, the most important two are the adjacency matrices and the Signless Laplacian matrices of graphs. This thesis mainly consider the adjacency matrices and Signless Laplacian matrices,mainly investigates the spectral radius and the adjacency matrices and Signless Laplacian matrices of non-regular connected graphs spectral radius, then try to build some connections between them and some structure variables of corresponding graphs. The main content of this thesis are as follows.(i)In Chapter1, we first look back the evolvement history of graph the-ory. And then introduce the backgrounds and research progresses of some questions on graph spectra theory that we study. At last, some definitions and notations for the corresponding questions are introduced. (ii) In Chapter2, we summarize some researchers’recent studying of the lower bound on the difference between the maximum degree and spectral radius which is Δ-λ of non-regular graph.(iii) In Chapter3, we discuss the connection between the maximum degree and spectral radius of κ—connected non-regular graphs, and get some new conclusions.(iv) In Chapter4, we investigate the connections between the Signless Laplacian spectral radius and the maximum degree, and get the lower bound on the difference between the twice maximum degree and Q—spectra radius.