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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.23773 |
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| _version_ | 1866917525027028992 |
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| author | Zhang, Jiechen |
| author_facet | Zhang, Jiechen |
| contents | We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Zhang, Jiechen Combinatorics Discrete Mathematics We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term. |
| title | A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2605.23773 |