On the minimum rank of a graph over finite fields

TL;DR

This paper investigates the minimum rank of graphs over finite fields, revealing asymptotic behavior over GF(2) and bounds related to clique size and field characteristics.

Contribution

It provides new bounds on the minimum rank over finite fields, especially for non-prime fields, and explores the average minimum rank over GF(2).

Findings

Average minimum rank over GF(2) tends to (1-ε)n with ε→0.

For non-prime q, minimum rank is at most n-k+1 when G contains a K_k.

Existence of graphs with minimum rank greater than 3 over GF(3) for certain parameters.

Abstract

In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on $n$ vertices over $\FF_2$ is $(1-\varepsilon_n)n$, were $\lim_{n\to\infty} \varepsilon_n=0$. In the second part of this paper we assume that $G$ contains a clique $K_k$ on $k$-vertices. We show that if $q$ is not a prime then $\mr(\FF_q,G)\le n-k+1$ for $4\le k\le n-1$ and $n\ge 5$. It is known that $\mr(\FF_q,G)\le 3$ for $k=n-2$, $n\ge 4$ and $q\ge 4$. We show that for $k=n-2$ and each $n\ge 10$ there exists a graph $G$ such that $\mr(\FF_3,G)>3$. For $k=n-3$, $n\ge 5$ and $q\ge 4$ we show that $\mr(\FF_q,G)\le 4$.