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

TL;DR

This paper derives formulas for specific Laplacian polynomial coefficients of forests, linking them to counts of closed walks in the forest and its line graph, extending previous work on tree invariants.

Contribution

It provides an exact formula for the coefficient c_{n-6} and expresses the coefficients c_{n-k} for 1 ≤ k ≤ 6 in terms of closed walks, advancing spectral graph theory.

Findings

Formulas for c_{n-6} in terms of closed walks.

Expressions for c_{n-k} (1 ≤ k ≤ 6) using closed walks.

Extended understanding of Laplacian coefficients for forests.

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 $F$, $1\leq k \leq 6$, are computed in terms of the number of closed walks in $F$ and its line graph.