Distance-regularised graphs are distance-regular or distance-biregular
The authors are concerned that the theory of distance-regular graphs does not apply directly to the graphs of generalised polygons. They therefore define a distance-regularised graph to be one in which, for any integer k and any vertices x and y, the number of vertices at distance k from x and adjacent to y only depends on the vertex x and the distance between x and y. It is then shown that te nonbipartite distance regularised graphs are distance regular, while the bipartite ones are distance-biregular. Much of the theory of distance-regular graphs is extended to distinct- regularised graphs. Apart from the generalised polygons, three classes of distance-biregular graphs are given. The question of whether these are all the distance biregular graphs remains open.
scientific article; zbMATH DE number 3997835