The Spectral Sufficient Conditions On Pancyclic Graph And Hamilton Graph With Toughness

It is an important NP-complete problem to judge whether a graph is Hamil-tonian.Since the spectrum of the graph can reflect the structural properties of the graph well and is easy to calculate.Therefore,using spectral graph theory to study the Hamiltonicity of graph has gradually become a research hotspot in the algebraic graph theory,and obtained a lot of good results in recent yearsPancyclic graph must be Hamilton graph,but there is much more complicated than Hamilton graph for researching.There have very few results for using spectrum of the graph characterization the pancyclicity of the graph.In this paper,we studies and presents the spectral sufficient conditions on pancyclic graph with the minimum degree is greater than or equal to 2 and greater than or equal to 3.In addition,all of Hamilton graph must be 1-toughness graph,toughness and Hamilton graph has a close relationship.In this paper,we first establish the spectral sufficient conditions for a graph to be Hamiltonian with toughnessThis paper is divided into four chapters,the detailed contents are as followsIn Chapter 1,firstly,we introduce the background and significance,then intro-duce the concepts,definitions and terminologies involved.Finally we introduce the progress and main conclusions in this paperIn Chapter 2,firstly,we give the sufficient condition of the edge number on Pancyclic graph with minimum degree greater than or equal to 2,and it is obtained by using the method of degree sequence.Then,we consider the relationship between edge number and extreme spectra.Finally,we give the spectral sufficient conditions on Pancyclic graph with minimum degree greater than or equal to 2 is obtained by using the spectral radius of the graph and the signless Laplacian spectral radius respectivelyIn Chapter 3,firstly,considering the influence of minimum degree condition of the graph,using the method is similar to chapter 2,and then we gives the edge sufficient condition of the pancyclic graph with minimum degree greater than or equal to 3.Finally,we gives the spectral sufficient conditions on pancyclic graph with minimum degree greater than or equal to 3In Chapter 4,firstly,we give the sufficient condition of the edge number on Hamilton graph with toughness,and it is obtained by using the degree sequence method of the graph.Then,we consider the relationship between edge number and extreme spectra.Finally,we give the spectral sufficient condition on Hamilton graph with toughness is obtained by using the spectral radius of the graph and the Signless Laplacian spectral radius respectively.