On the spectral moments of signless Laplacian matrix of trees and unicyclic graphs

Let Q(G) = D(G) + A(G) be the signless Laplacian matrix of G, where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of vertex degrees. Let q(1) (G), q(2) (G), ... , q(n-1) (G), q(n) (G) be the eigenvalues in non-increasing order of Q(G). The number Sigma(n)(i=1) q(i)(k) (G) (k = 0, 1, ... , n-1) is called the k-th signless Laplacian spectral moment of G, denoted by T-k(G), and T (G) = (T-0(G), T-1(G), ... , Tn-1(G)) is the sequence of signless Laplacian spectral moments of G. For two graphs G(1), G(2), we shall write G(1) <(T) G(2) if for some k(1 <= k <= n-1), T-i (G(1)) = (G(2)) (i = 0,1, ... , k-1) and T-k (G1) < T-k (G2) hold. In this paper, we prove that the T-order of a graph is monotonic on some transformations, and determine the first and the last graphs, in a T-order, of all trees and unicyclic graphs of order n, respectively, and finally characterize the extremal graphs for trees of order n with diameter d and unicyclic graphs of order n with girth g.