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Applied Mathematics 2025

The Laplacian Permanents and Laplacian Ratios of Trees

DOI: 10.4236/am.2025.164017, PP. 338-346

Xiangshuai Dong

Keywords: Laplacian Permanents, Laplacian Ratio, Tree, Caterpillar

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Abstract:

Brualdi and Goldwasser characterized the Laplacian permanents of trees. In this paper, we study the Laplacian permanents of trees. We characterize some Laplacian permanents of trees. The Laplacian ratio of $G$ is the Laplacian permanent of $G$ divided by the product of degrees of all vertices. In this paper, we obtain that for any $n$ -vertex caterpillar tree $T$ , there exists an $n$ -vertex caterpillar tree $C_{n}$ such that $π (C_{n}) \leq π (T)$ .

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