Study On Structural Parameters Of Graph And Properties Of Laplacian Eigenvalues

The subject of graph theory is a branch of applied mathematics,which is widely used in many fields.Scholars have studied and discussed graph theory from different perspectives.In order to study the properties of graphs,adjacency matrices and Laplace matrices are introduced.Both of these matrices are closely related to the structure of graphs.Compared with the eigenvalues of Laplacian matrix and adjacency matrix,the former is closely related to the properties and structure of graph,which reflects the graph theoretical properties of graph.Therefore,many academic researches focus on the eigenvalues of Laplacian matrix to carry out a series of discussions and analysis,and get rich research results.It is not difficult to see that the research on the eigenvalues of Laplacian matrix has become a hot topic in recent years.Scholars have carried out research and analysis from different perspectives and obtained many research results,which laid a theoretical foundation for the study of this paper.Let G be a connected graph of order n and mG(I)be the number of Laplacian eigenvalues of G in an interval I.If I={?)for a real number ?,then mG(?)is just the multiplicity of ? as a Laplacian eigenvalue of G.It is well known that the Laplacian eigenvalues of G are all in the interval[0,n].A lot of attention has been paid to the distribution of Laplacian eigenvalues in the smallest subinterval[0,1)of length 1 in[0,n].Particularly,Hedetniemi et al.have proved that if the domination number of G is ?,then mG[0,1)??.We are interested in another extreme problem:The distribution of Laplacian eigenvalues in the largest subinterval(n-1,n]of length 1.In the first chapter of this article,we first introduce some basic concepts of related problems,and then review the development of graph theory and introduce the background and progress of the problem studied in this article.In the second and third chapters,we mainly prove that mG(n-1,n]?? and mG(n-1,n]??-1,where ?and ? are respectively the vertex-connectivity and the chromatic number of G.Two other main results of this paper focus on mG(?),the multiplicity of an arbitrary Laplacian eigenvalue ? G.of G.In the fourth chapter,we mainly prove that mG(?)?n-?.In the fifth chapter,we mainly prove mG(?)??/?+1n and the equality of the inequality.