The Characterization Of Spectra And Eccentricity-type Topological Indices For Some Properties Of Graphs

Deciding whether a given graph has a certain property is always a hot topic in graph theory.Because the spectra of a graph can well reflect the structural properties of the graph and are easy to calculate,it is more and more common to study the structural properties of the graph by using the theory of the graph in recent years,and many important conclusions have been obtained.In chemical graph theory,we use molecular graph to describe the topological structure of chemical molecules.Topological indices are a kind of topological invariants in theoretical chemistry.They are real numbers independent of graph drawing and label,reflecting the structural properties of chemical molecules to a certain extent.So topological indices are also used to describe all kinds of properties of graphs.In this paper,some properties of graphs are described from different angles,including spectral radius,signless Laplacian spectral radius and three kinds of topological indices based on eccentricity.The specific arrangement is as follows:In chapter 1,firstly,the background and significance of this paper are introduced,and then the basic symbols and concepts of this paper are introduced.Finally,the main conclusions of this paper are expounded.In chapter 2,according to the different properties of graphs with corresponding stability,the closure operation is carried out and the corresponding closure is constructed.Then,the classification of the complementary graph of the closure is discussed.Finally,when the spectral radius of the complementary graph does not exceed a certain number,the graph given a larger minimum degree isα（G）≤s,s-Hamilton-connected or containsC_s,P _s,K _2,s,s K₂ and s-factors.In chapter 3,closure operation is performed on the original graph and corresponding closure is constructed,and then the classification of the complementary graph is discussed.Finally,when the signless Laplacian spectral radius of the complementary graph does not exceed a certain number,the graph isα（G）≤k,Hamilton-connected or contains C_s,P _s,K _2,s,s K₂ and s-factors.In chapter 4,we first give the degree sequence conditions or edge sufficient conditions for different properties of graphs,and then use algebraic operation and inequality correlation techniques to propose sufficient conditions for graphs to be weakly Hamilton-connected,β-deficient,or k-edge-connected according to the eccentric connectivity index,eccentric distance sum and connective eccentricity index respectively.In chapter 5,according to the research content of the first three chapters,the main research methods and conclusions of this paper are summarized,and the shortcomings of this paper are pointed out and the contents of the research can be studied.