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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2505.01633 |
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| _version_ | 1866916955918696448 |
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| author | Gharakhloo, Roozbeh Latimer, Tomas Lasic |
| author_facet | Gharakhloo, Roozbeh Latimer, Tomas Lasic |
| contents | Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2ν,j)$, of connected $2ν$-valent labeled graphs with $j$ vertices that can be embedded on a compact Riemann surface of minimal genus $g$. We also derive formulae for their two-legged counterparts $\mathcal{N}_{g}(2ν,j)$. Our method recovers the known explicit results for graphs embedded on the plane and the torus, and extends them to all genera $g \geq 2$. In earlier work, Ercolani, Lega, and Tippings (2023) showed that $\mathscr{N}_{g}(2ν,j)$ and $\mathcal{N}_{g}(2ν,j)$ admit structural expressions as linear combinations of, respectively, $3g-2$ and $3g$ Gauss hypergeometric functions ${}_2F_1$, but with coefficients left undetermined. The framework developed here provides a systematic procedure to compute these coefficients, thereby turning the structural expressions into fully explicit formulae for $\mathscr{N}_{g}(2ν,j)$ and $\mathcal{N}_{g}(2ν,j)$ as functions of both $j$ and $ν$. Detailed results are given for $g=2,3,$ and $4$, and the framework extends naturally to all $g \geq 5$ with increasing computational effort. This closes the fixed genus combinatorics for even-valent graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorics of Even-Valent Graphs on Riemann Surfaces Gharakhloo, Roozbeh Latimer, Tomas Lasic Combinatorics Mathematical Physics 5A15, 05C30, 39A60, 41A60, 33C45, 47E07 Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2ν,j)$, of connected $2ν$-valent labeled graphs with $j$ vertices that can be embedded on a compact Riemann surface of minimal genus $g$. We also derive formulae for their two-legged counterparts $\mathcal{N}_{g}(2ν,j)$. Our method recovers the known explicit results for graphs embedded on the plane and the torus, and extends them to all genera $g \geq 2$. In earlier work, Ercolani, Lega, and Tippings (2023) showed that $\mathscr{N}_{g}(2ν,j)$ and $\mathcal{N}_{g}(2ν,j)$ admit structural expressions as linear combinations of, respectively, $3g-2$ and $3g$ Gauss hypergeometric functions ${}_2F_1$, but with coefficients left undetermined. The framework developed here provides a systematic procedure to compute these coefficients, thereby turning the structural expressions into fully explicit formulae for $\mathscr{N}_{g}(2ν,j)$ and $\mathcal{N}_{g}(2ν,j)$ as functions of both $j$ and $ν$. Detailed results are given for $g=2,3,$ and $4$, and the framework extends naturally to all $g \geq 5$ with increasing computational effort. This closes the fixed genus combinatorics for even-valent graphs. |
| title | Combinatorics of Even-Valent Graphs on Riemann Surfaces |
| topic | Combinatorics Mathematical Physics 5A15, 05C30, 39A60, 41A60, 33C45, 47E07 |
| url | https://arxiv.org/abs/2505.01633 |