A characterization of the graphs of bilinear $(d\times d)$-forms over $\mathbb{F}_2$

TL;DR

This paper proves that the bilinear forms graph over the binary field for square matrices of size at least 3 is uniquely characterized by its intersection array, filling a gap in the classification of these graphs.

Contribution

It establishes the characterization of the bilinear forms graph over  for square matrices of size  or more, completing the classification for previously unsettled cases.

Findings

The graph of bilinear (d) forms over  is characterized by its intersection array for d .

Classifies locally grid graphs with hexagon -graphs and well-defined intersection numbers.

Fills the gap in the classification of bilinear forms graphs over  for certain matrix sizes.

Abstract

The bilinear forms graph denoted here by $Bil_q(e\times d)$ is a graph defined on the set of $(e\times d)$-matrices ($e\geq d$) over $\mathbb{F}_q$ with two matrices being adjacent if and only if the rank of their difference equals $1$. In 1999, K. Metsch showed that the bilinear forms graph $Bil_q(e\times d)$ is characterized by its intersection array if one of the following holds: (-) $q=2$ and $e\geq d+4$, (-) $q\geq 3$ and $e\geq d+3$. Thus, the following cases have been left unsettled: (-) $q=2$ and $e\in \{d,d+1,d+2,d+3\}$, (-) $q\geq 3$ and $e\in \{d,d+1,d+2\}$. In this work, we show that the graph of bilinear $(d\times d)$-forms over the binary field, where $d\geq 3$, is characterized by its intersection array. In doing so, we also classify locally grid graphs whose $\mu$-graphs are hexagons and the intersection numbers $b_i,c_i$ are well-defined for all $i=0,1,2$.