Several Problems Based On The Eigenvalues Of Graphs And Its Applications

The study of spectrum theory is an important branch of graph theory.The spectrum theory has very close relationships with the algebraic and topological properties of graphs,which aims to explore the structural properties of the corresponding graphs in terms of the algebraic methods and combinatorial techniques.Another interesting problem in this aspect is to investigate the relationship between the eigenvalues of various graph matrices and graph parameters(such as connectivity,chromatic number and diameter of graphs etc.).With the rapid development of information technology,the spectrum theory has been playing an important role in physics,chemistry,bioinformatics and computer science.Spectrum theory can illustrate efficiently the structural properties of graphs.Generally speaking,there are several kinds of spectrum of graphs including the spectrum of adjacent matrix,Laplacian spectrum and signless Laplacian spectrum.It was in 2017 that Professor V.Nikiforov proposed the concept of theα-matrix of graph G:_αA(G)=αD(G)+(1-α)A(G),where 0≤α≤1.It is routine to check that this new matrix is the linear convex combination of the corresponding adjacent matrix A(G)and the degree diagonal matrix D(G).In this paper,we mainly focus on several problems based on the eigenvalues of graphs and its applications.In the first chapter of the paper,we briefly introduce the development of graph theory,and also present the background and valuable applications of the spectrum theory.Some basic concepts and notations are also introduced in this part which will be used throughout of the paper.In Chapter 2,we introduce several properties of the bidegreed split graphs and combinatorial designs.In Chapter 3,we propose two necessary and sufficient conditions for connected bidegreed split graphs to have exactly 4 different eigenvalues.Our results generalize several well-known results which were proved by F.Goldberg and G.Su in the past two years.In the final Chapter of this thesis,we first summarize the main works and then propose several considerable problems which will be explored in the future.