Laplacian coefficient, matching polynomial and incidence energy of trees with described maximum degree

Let L(T,λ)=∑k=0n(-1)kck(T)λn-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}(T,\lambda )=\sum _{k=0}^n (-1)^{k}c_{k}(T)\lambda ^{n-k}$$\end{document} be the characteristic polynomial of its Laplacian matrix of a tree T. This paper studied some properties of the generating function of the coefficients sequence (c0,…,cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(c_0, \ldots , c_n)$$\end{document} which are related with the matching polynomials of division tree of T. These results, in turn, are used to characterize all extremal trees having the minimum Laplacian coefficient generation function and the minimum incidence energy of trees with described maximum degree, respectively.