Laplacian Spetral Radius And Signless Laplacian Spetral Radius Of Graphs

In graphs theory, people introduce various matrices, such as the adjacency matrix, the distance matrix, the Laplacian matrix, the signless Laplacian matrix, to research the properties of graphs by studying the algebraic properties of matrices.The matrices that people most usually study are adjacency matrix, Laplacian matrix and signless Laplacian matrix. Comparing with adjacency matrix, Laplacian matrix and signless Laplacian matrix contain the information about the degrees of all vertices. It can show the some properties of graphs well. This thesis will research the problem of Laplacian matrix. The main content can be divided into three chapters.The first chapter gives the background and development of Laplacian spectral radius and signless Laplacian spectral radius, introduces the related definitions and notations, and explains the structure of this thesis.The second chapter studies the effect on the signless Laplacian spectral radius under various perturbations, and studies the unicyclic graph with maximal signless Laplacian spectral radius.The third chapter studies the signless Laplacian spetral radius of unicyclic graphs with fixed order and girth, and find out the unicyclic graph (fixed order and girth) with maximal signless Laplacian spectral radius. It also studied the Laplacian spetral radius of unicyclic graphs(fixed order and girth is even)and find out unicyclic graphs(fixed order and girth is even)with maximal Laplacian spectral radius.