The Double Star Is Determined By Its Distance Spectrum

Graph theory originated in the early 18 th century, when the famous mathematician Euler proposed the seven bridge problem. The famous Euler’s formula presents the relation between the number of edges, vertices, and faces of a convex polyhedron. Graph theory has attracted wide attention as its extremely extensive application in chemistry,information system, network system and even in sociology, and the research of spectral graph theory is a hot branch of the research of Graph theory. Spectral graph theory mainly studies the combination properties of a graph by investigating the characteristic polynomial, eigenvalue and eigenvector of the corresponding graph matrix( such as adjacency matrix, Laplacian matrix, unsigned-Laplacian matrix and distance matrix matrix).This research has made a lot of beautiful results, for example, the connected graphs with exactly two different eigenvalues are complete graphs; the connected graphs with exactly one positive eigenvalue are complete multipartite graphs; the graphs with the smallest eigenvalue not less than-2 are line graphs, generated line graphs or finite number of special graphs; the graphs with spectrum containing opposite number of spectral radius are bipartite graphs; the graphs with spectrum containing all eigenvalues and their reciprocals are corona graphs and so on. When it comes to graph spectral theory, the famous mathematician N.W illiams points out that: “some important problem seems to be pure combination have such characteristics, i.e. it is impossible to get the existing conclusions without graph adjacency matrix eigenvalue algebraic method.”We call two graphs cospectral if they have the same adjacency spectrum. We call a graph determined by its spectral(or DS for short) if there exists no non-isomorphism graphs cospectral with it. The DS problem is an important aspect to study the spectral graph theory. The DS problem goes back for about half a century, and originates from chemistry. In recent decades, many mathematicians generalized the problem to Laplacian spectrum, signless Laplacian spectrum or generated spectrum. Recently, the distance spectra of graphs have received widespread attention, in this paper we showed that the double star is determined by its distance spectrum.This paper is organized as follows. In Chapter 1, we first introduce some background of spectral graph theory, the raise of the problem of determined by distance spectra and some relevant applications. Next we introduce some fundamental notions and symbols. At last we list some known results about determined by distance spectra. Chapter 2 contains two sections. In the first section, we introduce the definition of the distance equitable partition. In the second section, we present some uses of distance equitable partition. In Chapter 3, we first give some useful lemmas to get some forbidden subgraphs, then prove that the double star is determined by its distance spectrum.