Ordering By Spectral Moments Of Unicyclic And Bicyclic Graphs With Fixed Diameter

Let G be a graph with order n and size m.Ifm=n,G is unicyclic；and if m=n+1,then G is bicyclic.Unicyclic graphs and bicyclic graphs with order n and diameter d are denoted by U(n,d)and B(n,d),respectively.Let A(G)be the adjacency matrix of a graph G,its eigenvalues (?)i are the eigenvalues of graph G.The k-th spectral moment of G is denoted by Sk(G)=(?)ki(G)(k=0,1,…,n-1).Let s(G)=(So(G),S1(G),…,Sn-1(G))be the sequence of spectral moments of G.For two graphs G1,G2,if for some k(k=1,2,…,n-1),we have Si(G1)=Si(G2)(i=0:1,…,k-1)and Sk(G1)<Sk(G2),then we say G1(?)s.G2(Gl comes before G2in S-order). In the paper we determine unicyclic graphs and bicyclic graphs,in an s-order,among the set U(n,d)and B(n,d),respectively.This article is composed of four sections.Section1is the introduction and prepa-ration,in which we introduce some basic concepts and overview the main conclusions Obtained by Cvetkovic、Rowlinson.Wu.Liu and so on.In section2,we determined the last [(d+1)/2] graphs,for d=n-2;the last d-1graphs,for3≤d≤n-3,in an S-order,of all unicycyclic graphs in the set U(n,d).In section3,we determined the last [d/2]-1graphs,for n≥d+5,in an S-order,of all bicyclic graphs in the set B(n,d).Section4summarizes the paper and gives expectations in the future.