Unimodular graphs and Eisenstein sums

TL;DR

This paper introduces unimodular graphs over finite fields and valuation rings, computes their spectra using Eisenstein sums, and applies these results to estimate solutions to dot product equations and determine isoperimetric constants.

Contribution

It develops the spectral theory of unimodular and Platonic graphs over finite valuation rings, providing explicit spectra and improved bounds for isoperimetric constants.

Findings

Computed spectra of unimodular graphs using Eisenstein sums.

Derived estimates for solutions to restricted dot product equations.

Established exact and improved bounds for isoperimetric constants.

Abstract

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated to unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation $a\cdot b=r$ over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings - e.g., $\mathbb{Z}/(N)$. In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over $\mathbb{Z}/(N)$.