On expander Cayley graphs from Galois rings

TL;DR

This paper introduces new Cayley graphs over Galois rings, proving they are expanders and, in some cases, Ramanujan graphs, with detailed spectral analysis and novel estimation techniques.

Contribution

It presents the first construction of expander and Ramanujan Cayley graphs from Galois rings, expanding the class of known optimal expanders.

Findings

Cayley graphs over Galois rings are proven to be expanders.

Galois rings of characteristic 4 yield new infinite Ramanujan graphs.

Spectral properties of these graphs are analyzed using character sum estimates.

Abstract

In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley graphs from Galois rings of characteristic 4 form a new infinite family of Ramanujan graphs by an elementary eigenvalue estimation. Moreover some other spectral properties of our graphs are also discussed.