Laplacian coefficients of trees with a given bipartition

Let G be a graph of order n and the Laplacian characteristic polynomial of G. Zhou and Gutman [19] proved that among all trees of order n, the kth coefficient ck is largest when the tree is a path and is smallest for a star. In this paper, for two given positive integers p and q (p≤q), we characterize the trees with a given bipartition (p,q) which have the minimal and second minimal Laplacian coefficients.