Deterministic Completion of Rectangular Matrices Using Asymmetric Ramanujan Graphs: Exact and Stable Recovery

Graphs and Hermitian matrices: Exact interlacing

We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient: In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest eigenvalue of a graph. Keywords: extreme eigenvalues, tight interlacing, graph Laplacian, singular values, nonnegative matrix 1 Introduction Our notation is st...

Matrices and α-Stable Bipartite Graphs

A square (0, 1)-matrix X of order n ≥ 1 is called fully indecomposable if there exists no integer k with 1 ≤ k ≤ n− 1, such that X has a k by n− k zero submatrix. The reduced adjacency matrix of a bipartite graph G = (A,B,E) (having A ∪ B = {a1, ..., am} ∪ {b1, ..., bn} as vertex set, and E as edge set), is X = [xij ], 1 ≤ i ≤ m, 1 ≤ j ≤ n, where xij = 1 if aibj ∈ E and xij = 0 otherwise. A sta...

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...

Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming

A partial pre-distance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidean distance matrix, EDM. The nearest (or approximate) Euclidean distance matrix problem is to find a Euclid...