Laplacian Coefficient, Matching Polynomial and Incidence Energy of of Trees with Described Maximum Degree

TL;DR

This paper investigates properties of Laplacian coefficients and matching polynomials of trees, characterizing extremal trees with minimal Laplacian coefficient generation functions and incidence energy based on maximum degree.

Contribution

It introduces new characterizations of extremal trees with minimal Laplacian coefficients and incidence energy using generating functions and matching polynomials.

Findings

Identified trees with minimum Laplacian coefficient generation function.

Characterized trees with minimum incidence energy for given maximum degree.

Established relationships between Laplacian coefficients and matching polynomials.

Abstract

Let $\mathcal{L}(T,\lambda)=\sum_{k=0}^n(-1)^{k}c_{k}(T)\lambda^{n-k}$ 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 $(c_0, \cdots, c_n)$ 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.