Ranks of matrices with few distinct entries

TL;DR

This paper investigates the maximum size of matrices with limited off-diagonal entries and bounded rank, providing classifications and bounds that have implications in extremal combinatorics and eigenvalue multiplicities.

Contribution

It classifies sets L for which N(r,L) is linear and establishes quadratic lower bounds for superlinear cases when L is a subset of integers.

Findings

Classified sets L with linear N(r,L)

Proved N(r,L) is at least quadratic for superlinear cases

Asymptotically determined maximum eigenvalue multiplicity in digraph adjacency matrices

Abstract

An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix of rank at most $r$. Many applications of linear algebra in extremal combinatorics involve a bound on $N(r,L)$. We review some of these applications, and prove several new results on $N(r,L)$. In particular, we classify the sets $L$ for which $N(r,L)$ is linear, and show that if $N(r,L)$ is superlinear and $L\subset \mathbb{Z}$, then $N(r,L)$ is at least quadratic. As a by-product of the work, we asymptotically determine the maximum multiplicity of an eigenvalue $\lambda$ in an adjacency matrix of a digraph of a given size.