Laplacian coefficients of trees with a given bipartition

Abstract

Let G be a graph of order n and μ ( G , λ ) = ∑ k = 0 n ( - 1 ) k c k λ n - k the Laplacian characteristic polynomial of G . Zhou and Gutman [19] proved that among all trees of order n , the k th coefficient c k 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.