Incidence Gain Graphs and Generalized Quadrangles

TL;DR

This paper introduces a new construction method for generalized quadrangles using gain functions on incidence graphs, applicable to both finite and infinite geometries, with explicit examples on affine planes.

Contribution

It presents a novel approach to constructing generalized quadrangles via gain functions on incidence graphs, extending to infinite geometries and providing explicit examples.

Findings

Constructed generalized quadrangles from affine planes using gain functions.

The method applies to both finite and infinite geometries.

Provided explicit examples on affine planes over arbitrary fields.

Abstract

We demonstrate a construction method based on a gain function that is defined on the incidence graph of an incidence geometry. Restricting to when the incidence geometry is a linear space, we show that the construction yields a generalized quadrangle provided that the gain function satisfies a certain bijective property. Our method is valid for finite and infinite geometries. We produce a family of generalized quadrangles by defining such a gain function on an affine plane over an arbitrary field.