On expander Cayley graphs from Galois rings

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.

A note on the relation between two properties of random graphs

“…At present, various such families have been obtained (see e.g. [2], [22]). On the other hand, almost no results on the converse direction have been obtained.…”