The Signless Laplacian Spectral Radius Of Graphs Without A Given Minor

Graph theory is a young and active branch of mathematics that currently has a wide range of applications in biology,medicine,computer,chemistry and physics.Algebraic graph theory is an important research direction in graph theory,which mainly studies graphs through the algebraic properties of associated matrices,and spectral graph theory mainly studies how the important properties and structures of graphs are reflected by the eigenvalues and eigenvectors of associated matrices.The graph associated matrices commonly studied include adjacency matrix,signless Laplacian matrix,Aα-matrix,Laplacian matrix,distance matrix and so on.In spectral extremal graph theory,the determination of spectral radius and the characterization of extremal graphs have formed a very popular and meaningful section.The well-known BrualdiSolheid-Turán type problem asks what is the maximum spectral radius of an H-free graph of order n,and characterize the extremal graph.The signless Laplacian matrix Q(G)of a graph G is defined as the sum of the diagonal matrix of vertex degrees and the adjacency matrix,and the largest eigenvalue of Q(G)is called the signless Laplacian spectral radius of G.The main work of this thesis is as follows:First,we introduce the origin and development of graph theory and spectral graph theory.Next,we review the research progress of the spectral extrema problem of graphs without a given minor.Finally,we investigate a signless Laplacian spectral radius version of Brualdi-Solheid-Turán type problem over all n-vertex K1,t-minor free graphs.We characterize the unique extremal graph with the maximum signless Laplacian spectral radius among all n-vertex connected K1,t-minor free graphs.For graphs that do not emphasize connectivity,we obtain the tight upper bound of the signless Laplacian spectral radius of n-vertex K1,t-minor free graphs,and identify the extremal graphs.