| The study of graph spectrum theory is one of the most important areas in graph theory research.It mainly uses the graph parameters described by the graph’s related matrix to describe the structural properties of the graph itself,and studies the internal connection between the graph’s topological parameters and its structure.The edge ex-pansion of the graph is to increase the number of edges of the graph G in some way to form a new graph.Many researchers have exploited the characterization of the Laplacian eigenvalues of edge expansion graphs.This thesis further extends the previous results.It mainly studies the spectrum of the clique expansion graph,the generalized quadrilateral graph and the graph obtained from the strong product.The specific content includes:In Chapter 1,we introduce the research background,list necessary notions and state the main results.In Chapter 2,we give the normalized Laplacian eigenvalues of the clique expansion graph CL(G).With the eigenvalues,we calculate the multiplicative degree-Kirchhoff index,the Kemeny’s constant and the number of spanning trees of CL(G).We also give the Laplacian eigenvalues of the graph obtained after r times.In Chapter 3,we give the normalized Laplacian eigenvalues of the generalized quadri-lateral graph Q(t)(G).And then calculate the multiplicative degree-Kirchhoff index,the Kemeny’s constant and the number of spanning trees of Q(t)(G),we also give the Lapla-cian eigenvalues of the graph obtained after r times.In chapter 4,we give the Laplacian eigenvalues and normalized Laplacian eigenvalues of the graph obtained from the strong product of the graph G and K2,and study the related parameters. |
