Research On The Distance Spectral Of Some Graphs

Graph and spectral theory occupies a very important position in graph theory research,and is mainly used in computer science,statistical mechanics,quantum chemistry,communication networks and other fields.Graph and spectral theory is mainly represented by algebra such as Distance matrix D(G),Adjacency matrix A(G),Laplacian matrix L(G),and clever application of matrix theory,linear algebra and other theories and skills to study the topological properties and certainty of graphs.It is a topic that has attracted the attention of many scholars in algep-braic graph theory,combinatorial matrix theory,and algebraic combinatorics,which greatly promotes and enriches the research content of graph theory and combinatorics.In this article,we mainly study the spectral of the distance matrix of the graph,referred to as the distance spectral of the graph.This paper mainly studies the distance spectral of graphs.Firstly,it introduces the re-search background and current status of the graph and spectral theory,and explains some basic concepts and basic symbols of the graph and spectral theory.Then two main research results obtained in this paper are introduced.One is based on the main sub-matrix of the graph,using Cauchy’s alternating theorem to describe all graphs with four negative distance eigenvalues;the other is based on the current research status of Turan-type problems,then the problem of minimum distance spectral radius is studied,and minimum distance spectral radius and cor-responding extremal graph of connected bipartite graph G is given when G does not contain two disjoint C4-cycles as subgraphs.That is to say,the sufficient conditions for all connected bipartite graphs containing two disjoint C4-cycles are given.Finally,the article is summarized,and the direction of future research on this subject is discussed.