New and Explicit Constructions of Unbalanced Ramanujan Bipartite Graphs

TL;DR

This paper introduces explicit methods for constructing unbalanced Ramanujan bipartite graphs, revisits existing construction techniques, and explores the possibility of creating such graphs with specific degree and edge restrictions.

Contribution

It provides the first explicit constructions of infinite families of unbalanced Ramanujan bigraphs and addresses the construction under edge restrictions.

Findings

Explicit constructions of unbalanced Ramanujan bigraphs are achieved.

Analysis of computational complexity for existing construction methods.

Feasibility of constructing Ramanujan graphs with specified degrees and limited prohibited edges.

Abstract

The objectives of this article are three-fold. Firstly, we present for the first time explicit constructions of an infinite family of \textit{unbalanced} Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of "prohibited" edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.