| Spectral Graph theory is an important research field of Algebraic Graph Theory and Combinatorial Matrix Theory.The essence of graph theory is to express graph as various kinds of matrices,such as Adjacency Matrix,Laplacian Matrix and so on,and then further reveal the structural properties of graph by analyzing matrix eigenvalues(also known as spectrum).Among them,the important research directions favored and concerned by experts majoring in graph theory and algebra are:1.Depicting the upper and lower bounds of the spectral radius of graphs and the corresponding extremal graphs;2.The multiplicity of each eigenvalue(nullity,some integer,etc.)and the rank of graphs;3.The energy of graphs.These problems have a series of practical applications in quantum chemistry,theoretical chemistry and information science.In this paper,we mainly study the Laplacian matrices and A_?matrices of simple graphs and the multiplicity of several kinds of eigenvalues of matrices.It is noted that these two matrices are linear combinations of adjacency matrix and degree matrix of graph,and they do not contain each other,but some properties are similar which arouse our interest.The full text is divided into four chapters:In Chapter 1,we mainly introduces the research background,significance,re-search status of these two types of matrix,and some basic concepts.In Chapter 2,we study the multiplicity of eigenvalues two of Laplacian matrices of bicyclic graphs with perfect matchings.In particular,we characterize the extremal bicyclic graphs containing perfect matchings of eigenvalue two of Laplacian matrices with multiplicity 3.In Chapter 3,we give the upper bound of the multiplicity of A_?matrix eigenvalues?of graphs with pendent numbers,and the upper and lower bounds of multiplicity of A_?matrix eigenvalues k?of graphs are also given.In Chapter 4,we summarizes the main work of this paper and introduces the work that can be further studied.There are totaly 2 figures and 95 references in this paper. |
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