| Determining whether the graph is Hamiltonian is a NP-complete problem.We are often used to use the spectrum theory of graphs to study the related structural properties of graphs.It is because the spectrum of graphs is convenient to be calculated,and then turn verification into calculation.It provides an easy and efficient determination method.So we use the spectrum theory of graphs to study the Hamiltonian properties of graphs.And it has become a hot issue.At the same time,many good conclusions have been given.This thesis mainly studies the traceability,Hamiltonicity and pancyclicity of graphs.The specific content is arranged as follows.In Chapter 1,firstly,the background and function of this thesis are described in detail,and then the relevant concepts,definitions and terms that often appear in this thesis are given.Finally,the research status of the Hamiltonian of graphs and the main conclusions of this thesis are briefly introduced.In Chapter 2,firstly,the edge condition for the graph to be traceable is optimized,and then we characterize the traceability of the graph in terms of the spectral radius and the signless Laplacian spectral radius,respectively.In Chapter 3,according to the relationship between the number of edges and the extreme spectrum,the Hamiltonian of graphs is characterized by the spectral radius of graphs or complements,and the signless Laplacian spectral radius,respectively.In Chapter 4,firstly,the edge condition for the graph to be pancyclic is optimized,and then we characterize the pancyclicity of the graph in terms of the spectral radius and the signless Laplacian spectral radius,respectively. |
