Ordering By Spectral Moments Of Quasi-trees

A connected graph G=(V,E)of order n is called a quasi-tree,if there exists u0∈V(G)such that G-u0is a tree.Denote L(n,do)={G:G is a quasi-tree graph of order n with G-u0being a tree and dG(u0)=do}.LetA(G)be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G)be the eigenvalues in non-increasing order of a graph G.The number∑n i=1λik(G)(k=0,1,…,n-1)is called the k-th spectral moment of G,denote 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,G2of order n,then we say G1(?)s G2(G1comes before G2in an S-order) if exist some k(k=1,2,…,n-1), we have Si(G1))=Si(G2)(i=0,1,…,k-1)and Sk(G1))<Sk(G2).In this paper,we determine the last three quasi-trees,in an S-order,among the set (?)(n,d0), respectively.This paper consists of three sections.In section one,we first introduced some basic concepts and marks and then overview of the main conclusions obtained by Cvetkovic, Rowlinson,Wu,Fan and Liu.In section two,we determine the last three quasi-trees, in an S-order,among the set (?)(n,d0)by using some efective methods,respectively. Section three summarizes the paper and gives expectations in the future.