Ordering Trees By The Laplacian Coefficient

There are many methods to order trees.For example,trees can be ordered by their eigenvalues.Grone and Merris used algebraic connectivity to order trees.The Wiener index of a n-vertex tree T is epual to the(n-2)nd Laplacian coefficient,W(T)=C_n-2(T).Ordering of trees based on their Wiener index had a long history.So trees of the same Wiener index may be ordered by other Laplacian coefficients.This paper is motivated by Mohar's paper which proposed"how to order trees by the Laplacian coefficients."We first investigate a total ordering of trees with diameters 3 and 4 by the Laplacian coefficients.Then these results are used to determine first several orderings of trees by the Laplacian coefficients.