Laplacian coefficients of trees with given number of leaves or vertices of degree two

Let G be a simple undirected n-vertex graph with the characteristic polynomial of its Laplacian matrix L(G), det(lambda I - L(G)) = Sigma(n)(k=0)(-1)(k)c(k)lambda(n-k). It is well known that for trees the Laplacian coefficient c(n-2) is equal to the Wiener index of G, while c(n-3) is equal to the modified hyper-Wiener index of graph. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize the trees with k leaves (pendent vertices) which simultaneously minimize all Laplacian coefficients. In particular, this extremal balanced starlike tree S(n, k) minimizes the Wiener index, the modified hyper-Wiener index and recently introduced Laplacian-like energy. We prove that graph S(n, n - 1 - p) has minimal Laplacian coefficients among n-vertex trees with p vertices of degree two. In conclusion, we illustrate on examples of these spectrum-based invariants that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution, and pose a conjecture on extremal unicyclic graphs with k leaves. (C) 2009 Elsevier Inc. All rights reserved.