On Subgraphs in Distance-Regular Graphs

Terwilliger [15] has given the diameter bound d < (s 1)(k 1) + 1 for distance-regular graphs with girth 2s and valency k. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd graphs and proves this in the case of girth 4. We show that the only distance-regular subgraphs of a hypercube with girth 6 are the doubled Odd graphs. If the girth is equal to 8, then its valency is at most 12.