On Delorme's construction of distance biregular graphs

Distance biregular graphs are a generalization of biparte distance regular graphs in the sense that the parameters with respect a vertex may depend on the bipartite half to which it belongs. In 1994, C. Delorme gave constructions of distance biregular graphs whose sets of vertices of a bipartite half is the set of points of the 3-dimensional affine space over a finite field. The other half can be described as the set of hyperplanes of the affine space, with some parallel classes deleted. Then the incidence graph of this incidence structure is a distance biregular graph. In this talk, we give a characterization of the set of parallel classes of hyperplanes to be deleted, in order for the resulting incidence graph is distance biregular (and further, distance regular). We also discuss possible generalization of this construction.