On the Spectral Moment of Quasi-trees and Quasi-unicyclic Graphs.

A connected graph G = (V, E) is called a quasi-tree ( resp. a quasi-unicyclic graph), if there exists u(0) is an element of V (G) such that G - u(0) is a tree(resp. a unicyclic graph). Let lambda(1) (G), lambda(2)(G), ... , lambda(n) (G) be the eigenvalues of the adjacency matrix A(G) of G and S(G) = (S-0 (G), S-1(G), ... Sn-1(G)) the sequence of spectral moments of a graph G, where S-k (G) = Sigma(n)(i=1) lambda(k)(i) (G) (k = 0,1,..., n - 1) is the k-th spectral moment of G. For two graphs G(1), G(2), we have G(1) < s G(2), if for some k (k = 1, 2,...,n - 1), we have S-i(G(1)) = S-i(G(2)) (i = 0, 1,..., k - 1) and S-k(G(1)) < S-k(G(2)). In [16], Pan, Liu and Liu first determined the last and the second last quasi-tree, in an S-order, in the set of all quasi-tree with n vertices. Motivated by this paper, we focus on this problem, and present a new approach, with a shorter proof, to determine the last graphs in the S-order in all the quasi-trees. Furthermore, we identify the last graph in the set of quasi-unicyclic graphs, in an S-order, by this approach.