Research On The Structure And Parameters Of Graphs Based On The Signless Laplacian Spectra And The Distance Spectra

The theory of graph spectra can usually be considered as an attempt to utilize the Linear Algebra including the well-developed theory of matrices,for the purpose of graph theory and its applications.In this thesis we consider the spectra of the signless Laplacian matrix(resp.the distance matrix)of some types of graphs,which includes the k-trees and the Halin graphs(resp.the cacti and the graphs with given connectivity).Our main results are the following:· Denote the set of all n-vertex k-trees by Jnk.The graphs among Jnk having the first(resp.the second and the third)largest signless Laplacian index q1(G)are characterized and the sharp upper bound on the signless Laplacian index of graphs in Jnk is determined.Furthermore,sharp upper bound on q1(G)(resp.q1(G)+ k,q1(G)· k,q1(G)-k,q1(G)/k)of graph G in ?k=1n-1 Jnk are determined.The corresponding extremal graphs are characterized respectively as well.· Denote the set of all n-vertex Halin graphs by gn.Sharp upper and lower bounds on the signless Laplacian indices of graphs among gn are determined and the extremal graphs are identified,respectively.As well,graphs in gn having the second and third largest signless Laplacian indices are determined,respectively.· We characterize n-vertex cyclic cactus with given matching number m which minimizes the distance spectral radius.As a consequence,we characterize n-vertex cyclic cactus with a perfect matching which minimizes the distance spectral radius.· We characterize the graphs of order n with given connectivity and diameter having minimum distance spectral radius.In addition,we determine the mini-mum distance spectral radius of graphs among the n-vertex graphs with given connectivity and independence number,and characterize the corresponding extremal graph,thus determining the minimum distance spectral radius of graphs among n-vertex graphs with given connectivity(resp.independence)number.