| The Laplacian coefficients of trees have been widely studied. For example:Zhou and Gutman proved that if T is a tree of order n, then ci(T)=mi(S(T)), for0≤i≤n, where the subdivision graph S(T) of T is obtained by inserting a new vertex of degree2for each edge of T and mi(S(T)) is the number of matchings of S(T) containing exactly i edges(shortly i-matchings). Ordering trees with diameter3and diameter4by the Laplacian coefficients have been already solved by Xiao-Dong Zhang. He showed that if T(p,q) is a tree of order n with diameter3, then for i=0,…,n,ci(T(p,q)) is an increasing function of p for1≤p≤[n/2]-1, i.e., T(1,n-3)≤T(2,n-4)≤…≤T([n/2]-1,[n/2]-1). He also proved that if T(n, q,k,p1,…,pk) is a tree of order n≥5with diameter4, then T(n,n-3,2,1,1)≤T(n, q, k,p1,…,pk), with equality if and only if T(n, q, k,p1,…,pk)=T(n, n-3,2,1,1). In this paper, we investigate a partial ordering of trees with diameter5by the Laplacian coefficients. These results are used to determine several orderings of trees with diameter5by the Laplacian coefficients.Let G=(V, E) be a simple graph with vertex set V and edge set E. If A(G) and D(G) are the adjacency matrix and degree diagonal matrix, respectively, then Q(G)=D(G)+A(G) is called the signless Laplacian matrix of a graph G. Moreover, the eigenvalues of Q(G) is called the signless Laplacian eigenvalues of G. Let ki(G) be the ith largest eigenvalue of the signless Laplacian of G and di(G) be the ith largest degree of G. Cvetovic proved that if G is a connected graph with order n>4, then k1(G)≥d1(G)+1with equality iff G is the star K1,n-1. Das showed that k2(G)≥d2(G)-1. Jianfeng Wang obtained that k3(G)≥d3(G)-1/22. Later in this paper, a lower bound for the fourth largest eigenvalue of the signless Laplacian of a graph is given in terms of the fourth largest degree of the graph. |
