New and Explicit Constructions of Unbalanced Ramanujan Bipartite Graphs
This provides foundational tools for constructing expander graphs with specific properties, which are crucial for applications in computer science, coding theory, and cryptography.
The paper presents the first explicit constructions of infinite families of unbalanced Ramanujan bipartite graphs, addresses computational implementation challenges of known construction methods, and shows that bipartite Ramanujan graphs with specified degrees can be constructed while avoiding prohibited edges in many cases.
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.