Laplacian Coefficients of a Forest in terms of the Number of Closed Walks in the Forest and its Line Graph

Abstract

Let $G$ be a finite simple graph with Laplacian polynomial $\psi (G,\lambda )={\sum }_{k=0}^n(-1{)}^{n-k}{c}_k{\lambda }^k$. In an earlier paper, the coefficients $c_{n-4}$ and $c_{n-5}$ for tree with respect to some degree-based graph invariants were computed. The aim of this paper is to continue this work by giving an exact formula for the coefficients $c_{n-6}$. As a consequence of this work, the Laplacian coefficients $c_{n-k}$ of a forest $F$, $1\le k\le 6$, are computed in terms of the number of closed walks in $F$ and its line graph.