The Number Of Cut Vertices And The Spectral Radii Of Graphs

The main problem in spectral graph theory is to discuss whether or how the structural property of a graph is characterized by its spectral property. Here, the spectral property mainly means those of the spectral radius and the least eigenvalue of related matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, etc.) of a graph. In past years, the adjacency spectral radius have received much attention, and has been a hot topic in spectral graph theory.This thesis mainly discussed the relationship between the number of cut ver-tices and the spectral radius of a graph, determines the unicyclic graphs with maxi-mum spectral radius or the second largest spectral radius among all unicyclic graph-s with given the number of cut vertices, and determines the graph with maximum spectral radius among all bicyclic graphs with given the number of cut vertices.This thesis is organized as follows. In Chapter one, we introduce a brief back-ground of the spectral graph theory, some concepts and notations, the problem and its development, and the results we obtained in this thesis.In Chapter two, we discuss the spectral radius of unicyclic graphs with given the number of cut vertices, and characterize the structure of the graph with max-imum spectral radius and the graph with second largest spectral radius among all unicyclic graphs with given the number of cut vertices. In the last chapter, we dis-cuss the spectral radius of bicyclic graphs with given the number of cut vertices, and determine the graph with maximum spectral radius among all bicyclic graphs with given the number of cut vertices.