On The Extremum Of Eigenvalues Of Two Kinds Of Matrices

Graph theory is an important research topic in algebraic graph theory.The essential properties of graphs are engraved by the spectral properties of graphs.Among them,the spectral properties of adjacency matrix,Laplace matrix and unsigned Laplace matrix are the hot issues in the study of graph theory,and their structural properties are mainly described by their corresponding matrices.In this paper,we mainly study the minimum eigenvalues of three circle graphs with n—4 suspension points and the extremum of the maximum unsigned Laplacian separation of three circle graphs and four circle graphs,and characterize their polar graphs accordinglyThe main contents of this paper are as followsThe first chapter,introduces the research background and significance of graph theory,some necessary symbols and basic concepts,as well as the research status of extremum in graph theory;In the second chapter,we study the minimum eigenvalues of the complement graph of a three cycle graph with order n and n-4 hanging points.On the basis of considering only simple undirected connected graph,the adjacency matrix of the graph is the matrix representing the adjacency relationship between vertices,and its minimum eigenvalues are the minimum eigenvalues of the graph.Starting from the structure of complement graph,this paper studies the essential properties of minimum eigenvalue of complement graph,The unique polar graphs with minimum eigenvalues of adjacency matrix in the class of complement graphs of three cycles with order n and n-4 suspension points are characterized;In the third chapter,we study the maximum unsigned Laplacian resolution of three cycle graphs and four cycle graphs.In this paper,Let G be a simple graph of order n,its unsigned Laplace eigenvalue is q₁（G）≥q₂（G）≥…≥ q_n（G）,and its unsigned Laplace resolution is S_Q（G）=q₁（G）-q₂（G）.Thus,we study the maximum unsigned Laplacian separation of three cycle graphs and four cycle graphs,and characterize their polar graphs accordingly.