The $\mathbb{F}_2$-Rank and Size of Graphs

TL;DR

This paper explores the structure and existence of certain graphs with minimal $\,\mathbb{F}_2$-rank, providing new combinatorial proofs, constructions for even $n$, and showing non-existence for odd $n$.

Contribution

It offers a new combinatorial proof of the existence and non-existence of graphs with specific $\,\mathbb{F}_2$-rank properties, introduces a related graph product, and constructs infinite families of such graphs.

Findings

Existence of graphs with rank $n$ for even $n$

Non-existence of twin-free graphs with minimal rank for odd $n$

Construction of strongly-regular quasi-random graphs with Hadamard matrices

Abstract

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.