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

Abstract

We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill–Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of these invariants is determined by the other two.

2000 AMS Subject Classification::

• 05C31
• 05C50
• 14T05
• 14H99

Keywords:

• graph Jacobians
• Tutte polynomials
• graph zeta functions
• tropical Brill-Noether theory
• Tutte’s rotor construction
• random graphs
• Cohen-Lenstra heuristics