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The objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of 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.

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In this paper we study the matrix completion problem: Suppose X ∈ R nr ×nc is unknown except for an upper bound r on its rank. By measuring a small number m ≪ n r n c of the elements of X, is it possible to recover X exactly, or at least, to construct a reasonable approximation of X? At present there are two approaches to choosing the sample set, namely probabilistic and deterministic. Probabilistic methods can guarantee the exact recovery of the unknown matrix, but only with high probability. At present there are very few deterministic methods, and they mostly apply only to square matrices. The focus in the present paper is on deterministic methods that work for rectangular as well as square matrices, and where possible, can guarantee exact recovery of the unknown matrix. We achieve this by choosing the elements to be sampled as the edge set of an asymmetric Ramanujan graph or Ramanujan bigraph. For such a measurement matrix, we (i) derive bounds on the error between a scaled version of the sampled matrix and unknown matrix; (ii) derive bounds on the recovery error when max norm minimization is used, and (iii) present suitable conditions under which the unknown matrix can be recovered exactly via nuclear norm minimization. In the process we streamline some existing proofs and improve upon them, and also make the results applicable to rectangular matrices.This raises two questions: (i) How can Ramanujan bigraphs be constructed? (ii) How close are the sufficient conditions derived in this paper to being necessary? Both questions are studied in a companion paper.

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