On the spectral moment of quasi-trees

Abstract

A connected graph G=(V,E) is called a quasi-tree, if there exists u0∈V(G) such that G-u0 is a tree. Denote Q(n,d0)={G:G is a quasi-tree graph of order n with G-u0 being a tree and dG(u0)=d0}. Let A(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(G). The number ∑i=1nλik(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 have G1≺SG2 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). In this paper, we determine the last and the second last quasi-tree, in an S-order, in the set Q(n,d0), respectively.