Distance regular covers of the complete graph

Distance regular graphs fall into three families: primitive, antipodal, and bipar-tite. Each antipodal distance regular graph is a covering graph of a smaller (usually primitive) distance regular graph; the antipodal distance graphs of diameter three are covers of the complete graph, and are the first non-trivial case. Many of the known examples are connected with geometric objects, such as projective planes and general&d quadrangles. We set up a classification scheme, and give new existence conditions and new constructions. A relationship with the theory of equi-isoclinic subspaces of KY " , as studied by Lemmens and Seidel, is investigated. A connected graph G is distance regular if for any two vertices, the number of vertices at distance i from the first and j from the second depends only on ij j, and the distance between the initial vertices. Since the two vertices may coincide, G is necessarily regular. Some examples are the cycle graphs, the Petersen graph and its line graph, and the skeletons of the Platonic solids. Distance regular graphs have important connections with other areas of combinatorics, including finite geometry and coding theory. For a general introduction to this area, see Biggs [23. A distance regular graph of diameter two is usually referred to as a strongly regular graph. Distance regular graphs are either primitive or imprimitive, and the imprimitive graphs are antipodal or bipartite (or possibly both). The imprimitive graphs are of special interest, as they give rise to smaller primitive graphs. In particular, the antipodal graphs are covering graphs of