A note on Jacobians, Tutte polynomials, and two-variable zeta functions of graphs

TL;DR

This paper investigates the relationships between Jacobians, Tutte polynomials, and two-variable zeta functions of graphs, providing examples that show these invariants are not mutually determined.

Contribution

It demonstrates through examples that Jacobians, Tutte polynomials, and zeta functions are independent invariants of graphs, addressing questions posed by Lorenzini.

Findings

Jacobians are not determined by Tutte polynomials or zeta functions

Tutte polynomials are independent of Jacobians and zeta functions

Two-variable zeta functions do not uniquely determine Jacobians or Tutte polynomials

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.