| Spectra of graphs mainly studies the relationship between the spectral properties of graph correlation matrices and the structural properties of graphs.In recent years,the study of the relation between the multiplicities of eigenvalues and structural parameters of graphs has become a popular topic of interest among scholars.This thesis investigates the relation between the multiplicity of-1 eigenvalue of the line graph of a unicyclic graph and the number of pendant vertices in the unicyclic graph,as well as the relation between the multiplicity of nonzero eigenvalue of trees and their matching number.The thesis is divided into four chapters.Chapter 1 introduces the research background,research status,and main research work of this thesis,as well as the necessary basic concepts and related symbols.In Chapter 2,we use the annihilator of a graph to study the relationship between the multiplicity of-1 eigenvalue of the line graph of a unicyclic graph and the number of pendant vertices in the unicyclic graph.We obtain an upper bound for the multiplicity of-1 eigenvalue and characterize the extreme graphs that achieve this upper bound.In Chapter 3,we study the relation between the multiplicity of nonzero eigenvalues of trees and their matching number.Using star complement technique,Rowlinson[1]proved that if a tree T,with diameter at least 4,has 1 as an eigenvalue of multiplicity k,then it has k+1 pendant edges that form an induced matching.We extend this result to any nonzero eigenvalue using the Parter-Wiener theorem.Furthermore,if μ≠0,we classify all trees T with mμ(T)=k and a maximum matching or a maximum induced matching containing k+1 edges,where mμ(T)represents the multiplicity of the eigenvalue μ of the tree T.In Chapter 4,we provide a summary of the research content and main conclusions of the thesis,as well as introducing related problem that can be further studied in the future. |
