Divisibility conditions for intersection numbers of certain bipartite distance-regular graphs

A connected graph is said to be distance-regular whenever given any two vertices x, y at path-length distance h apart, the number of vertices at distance i from x and j from y is a fixed constant (called an intersection number of the graph) that only depends on h, i, j, and not the vertices x, y. The classification of all distance-regular graphs of sufficiently large diameter is an open problem that, at least for now, seems out of reach. An active area of research is the classification of distance-regular graphs satisfying certain additional properties. This paper is motivated by a paper of Miklavic which found divisibility conditions on the intersection numbers of certain bipartite Q-polynomial distance-regular graphs. We generalize his work to show that the same divisibility conditions hold for a larger set of bipartite distance-regular graphs.