Some Spectral Characters Of Graphs

The theory of graphs spectra is a crossing area between graph theory and algebra. Itis also a branch of algebraic graph theory. This paper is focus on the adjacency spectra ofgraphs and the spectra of weighted double stars. The main results are as follows:In the first chapter, introduction, we give the basic definitions, symbols and notationsabout graphs spectra. We also give the definitions of weighted graphs, and list some impor-tant known results about the spectral radii of adjacency spectra and the spectral radii ofweighted trees. At last, we introduce the methods of comparing with characteristic polyno-mials and comparing with eigenvectors.The second chapter consists of three sections. In the first section, we characterize thestructure of graphs with maximum spectral radii where each graph has given number ofvertices |V| and edges |E|: i.e., the maximum graph G satisfies the maximal condition: or , for ; moreover, G has diameter nomore than 2 and has no induced cycle of length greater than 3, besides, the vertex set of G*can be divided into three parts: the vertex in center clique, the pendant vertex (may be (?))and the vertex in attachment (may be (?)). In the second section, we give a bound of spectralradii of graphs with given number of basic cycles a and the number of vertices sufficientlylarge. Let a = |E| - |V | + 1 be the number of basic cycles of graph G, B(n,a) be a familyof graphs in which each graph has n vertices and a basic cycles. Let be the graphs satisfies the maximal condition, if n > [a(a + 2)]2, we have for . In the third section, we determine the graphs Tn,d0 with maximal spectralradii in the quasi-unicycle graphs C (n,d0) (for the definitions of C (n,d0) and Tn,d0, seeSection 2.3).In the third chapter, we give the spectra of weighted double stars and determine theweighted double stars with maximal spectral radii. Let Sw(a,b) be a weighted double starwith weight W = {w_i|w_j≥w(_j+1) > 0, j = 1,2,···,m - 1; i = 1,2,···,m}, set s₁ = andthen its spectra areLet S^W(a,b) be the maximal weighted double star, then its weight is put as in Fig 5 (b).