Density of Zeros of the Tutte Polynomial

TL;DR

This paper proves that the zeros of the Tutte polynomial are densely distributed in specific regions of the plane, advancing understanding of their complex behavior and partially confirming a longstanding conjecture.

Contribution

It establishes the density of Tutte polynomial zeros in certain plane regions, providing the first such result for regions with positive volume.

Findings

Zeros form a dense subset in specific plane regions

First density result for Tutte polynomial zeros in positive volume regions

Almost confirms a conjecture of Jackson and Sokal

Abstract

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.