Jacobian graphs

TL;DR

This paper introduces jacobian graphs, a new family of regular graphs that mimic the spectral properties of random graphs while having distinct local structures, using advanced algebraic geometry and number theory techniques.

Contribution

The paper presents a novel construction of jacobian graphs leveraging generalized jacobians and equidistribution theorems, bridging algebraic geometry with graph theory.

Findings

Jacobian graphs are spectrally indistinguishable from random graphs.

They exhibit significantly different local structures compared to typical random graphs.

The construction uses properties of algebraic curves and character sums over finite fields.

Abstract

We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.