University of Kashan Mathematics Interdisciplinary Research 2538-3639 10 2 2025 06 01 Laplacian‎ ‎Coefficients of a‎ ‎Forest in Terms of the Number of Closed Walks in the Forest and its Line Graph 133 143 114890 10.22052/mir.2024.255007.1467 EN Ali Ghalavand ‎Department of Pure Mathematics, ‎Faculty of Mathematical Sciences, ‎University of Kashan, ‎Kashan‎, ‎I‎. ‎R‎. ‎Iran Alireza Ashrafi ‎Department of Pure Mathematics, ‎Faculty of Mathematical Sciences, ‎University of Kashan, ‎Kashan‎, ‎I‎. ‎R‎. ‎Iran Journal Article 2024 06 05 ‎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. https://mir.kashanu.ac.ir/article_114890_4086ca2b1bbd37b77e9cca3d4e7d2f73.pdf