Projective Rectangles: The Graph of Lines

TL;DR

This paper studies the properties of the graph of lines in projective rectangles, revealing their strong regularity, clique structure, and connections to bilinear forms graphs, with implications for graph properties and partial geometries.

Contribution

It establishes the relationship between projective rectangles and bilinear forms graphs, providing new representations and confirming the main construction's validity.

Findings

Graph of lines in projective rectangles is strongly regular.

The main construction produces valid projective rectangles.

Connections to bilinear forms graphs and partial geometries.

Abstract

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique structure. The main construction of projective rectangles, stated in a previous paper, gives rectangles whose graph of lines is a known strongly regular bilinear forms graph. That fact leads to a proof that the main construction does produce projective rectangles, and also gives a new representation of bilinear forms graphs. We conclude by mentioning a few simple graph properties, such as the chromatic number, which is not known, and a partial geometry obtained from the graph.