A Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs

DOI10.1080/10586458.2014.917443zbMath1310.05121arXiv1309.3340OpenAlexW1994060846MaRDI QIDQ5246355

Julien Clancy, Timothy Leake, Sam Payne

Publication date: 21 April 2015

Published in: Experimental Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1309.3340

zbMATH Keywords

random graphsTutte polynomialsCohen-Lenstra heuristicsgraph zeta functionstropical Brill-Noether theorygraph JacobiansTutte's rotor construction

Mathematics Subject Classification ID

Graph polynomials (05C31) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Other Dirichlet series and zeta functions (11M41) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10) Curves in algebraic geometry (14H99)

Related Items

The distribution of sandpile groups of random graphs ⋮ Sandpile groups and the coeulerian property for random directed graphs ⋮ On a Cohen-Lenstra heuristic for Jacobians of random graphs ⋮ Probability theory for random groups arising in number theory ⋮ Sandpile groups of supersingular isogeny graphs ⋮ Gluing of graphs and their Jacobians ⋮ Two-vertex generators of Jacobians of graphs ⋮ Chip-Firing Games and Critical Groups ⋮ Realization of groups with pairing as Jacobians of finite graphs ⋮ The distribution of sandpile groups of random regular graphs ⋮ Moments and interpretations of the Cohen-Lenstra-Martinet heuristics ⋮ Divisors on matroids and their volumes ⋮ Divisors on matroids and their volumes ⋮ Smith normal form in combinatorics ⋮ Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs

Cites Work

• A tropical proof of the Brill-Noether theorem
• Specialization of linear systems from curves to graphs (with an appendix by Brian Conrad)
• The rotor effect can alter the chromatic polynomial
• Chip firing and the Tutte polynomial
• On the sandpile group of dual graphs
• The Tutte polynomial as a growth function
• Contraction-deletion invariants for graphs
• On a two-variable zeta function for number fields.
• Codichromatic graphs
• Riemann-Roch and Abel-Jacobi theory on a finite graph
• The monodromy pairing and discrete logarithm on the Jacobian of finite graphs
• The critical group from a cryptographic perspective