Explicit construction of Ramanujan bigraphs

TL;DR

This paper presents an explicit method to construct an infinite family of bipartite, biregular Ramanujan graphs using quotients of Bruhat-Tits buildings related to $SU_3(\

Contribution

It introduces a new explicit construction of Ramanujan graphs based on algebraic groups and their associated buildings, expanding known examples.

Findings

Constructed an infinite family of Ramanujan graphs

Graphs are bipartite and biregular

Method uses quotients of Bruhat-Tits buildings

Abstract

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of $SU_3(\mathbb Q_p)$. To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.