Gonality of Expander Graphs

TL;DR

This paper establishes lower bounds on the gonality of graphs based on spectral and expansion properties, showing that random 3-regular graphs typically have gonality exceeding one seventh of their genus.

Contribution

It introduces new bounds linking gonality with spectral and edge expansion, and applies these to analyze random 3-regular graphs.

Findings

Gonality is bounded below by spectral and expansion parameters.

Random 3-regular graphs have gonality > 1/7 of their genus asymptotically almost surely.

Abstract

We provide lower bounds on the gonality of a graph in terms of its spectral and edge expansion. As a consequence, we see that the gonality of a random 3-regular graph is asymptotically almost surely greater than one seventh its genus.