The Spectral Moment Of Graphs With Given Clique Number Or Chromatic Number

Given an n-vertex graph G, let A(G) be the adjacency matrix of G with λ1(G), λ2(G),...,λn(G) being its eigenvalues in non-increasing order. The number Sk(G)=∑ni=1λki(G)(k=0,1,..., n-1) is called the kth spectral moment of G, denoted by Sk(G). Let S(G)=(S0(G),S1(G),...)Sn-1(G)) be the sequence of spectral moments of G. For two graphs G1, G2, we shall write G1=s G2if Si(G1)=Si(G2) for i=0,1,...,n-1. Similarly, we have G1-<s G2(G1comes before G2in the S-order) if for some k (1≤k≤n-1), we have Si(G1)=Si(G2)(i=0,1,..., k-1) and Sk(G1)<Sk(G2). We shall also write G1≤G2if G1≤s G2or G1=s G2. This paper which stands on the basis of previous results, do further research on the spectral moment of connected graphs with given clique number or chromatic number. The concrete content is in the following:●In the first two chapters, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound dis-cussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation.●In Chapter3, we study the spectral moment of all n-vertex connected graphs with given clique number t (3≤t≤n-1).●In Chapter4, we study the spectral moment of all n-vertex connected graphs with given chromatic number x (3≤X≤n-1).●In Chapter5, we give the main conclusion of the article.