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

Abstract

‎In this paper‎, ‎we deal with calculating the laplacian coefficients of a finite simple graph $G$ with the Laplacian polynomial $\psi(G,\lambda) = \sum_{k=0}^{n}(-1)^{n-k}c_k\lambda^k$‎. ‎We also explore the relationship between the number of closed walks in a graph and a series of its line graphs with the Laplacian coefficients‎. ‎Our objective is to find a way to determine the Laplacian coefficients using the number of closed walks in a graph and its line graph‎. ‎Specifically‎, ‎we have derived the Laplacian coefficients $c_{n-k}$ of a forest $F$ (where $1 \leq k \leq 6$) in terms of the number of closed walks in $F$ and its line graph‎.

Keywords

• forest‎
• ‎laplacian coefficient‎
• ‎closed walk