Traceability And Hamilton-Connectivity Spectral Characterization Of Graphs

Spectral graph theory is an important research direction of algebr--aic graph theory.It is also an important topic in the study of combinatorial matrix theory and algebraic combination at home and abroad.It has a wide range of applications in other non-mathematical disciplines such as chemistry,statistical mechanics,communication networks computers and information science.Spectral graph theory is the use of matrix theory to relate the properties of the basic structure of a graph with the parameters of the correlation matrix of the graph,and to find out the intrinsic link between them.In order to study how the properties of graphs are related to the matrix of these graphs reflected by the nature of algebra,people have introduced a variety of different matrices,such as: adjacency matrices A(G),association matrices M(G),distance matrices D(G),Laplacian matrices L(G),and signless Laplacian matrices Q(G)of graphs.The Hamiltonian problem is to determine whether a given simple graph contains a Hamiltonian circle.It is a classic problem of graph theory research.To date,although there have been many related research findings,the most ideal solution has not yet been found.In recent years,people have begun to apply spectral theory to the study of this problem.Since Fiedler and Nikiforov characterized the Hamiltonian using the spectral radius of the graph's adjacency matrix in 2010,many graphiologists have been working on the study of the Hamiltonian utilize the spectrum of the graph and has drawn many good conclusions.The main content of this article is as follows,In the first chapter,we introduce the research background and significance of spectral graph theory,as well as the academic terminology and basic concepts involved in this paper.Then we describe the research problems and related progress,and show the main conclusions of this paper;In chapter 2,Spectral Radius Condition of Traceable Graphs;In chapter 3,Laplacian Spectral Sufficient Conditions for Hamilton-Connected Graphs and Traceable Graphs.