Incidence Gain Graphs and Generalized Quadrangles

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.