Laplacian coefficients of trees

Abstract

Let  G  be  a  simple  and  undirected  graph  with  Laplacian  polynomial  ψ(G, λ)  =  Σ_k=0ⁿ (-1)^n-kc_k(G)λ^k. In this paper, exact formulas for the coefficient c_n-4 and the number of 4-matchings with respect to the Zagreb indices of a given tree are presented. The chemical trees with first through the fifteenth greatest c_n-4-values are also determined.