Bipartite regular graphs with fixed diameter

For given nonnegative integers k and D , we consider the problem of determining n 0 (k, D), The smallest number n for which there exists a k ‐regular bipartite graph on n vertices with diameter D . We solve the problem for all pairs (k, D ) with D ≢ 2 (mod 4) and D ≢ 3 (mod 4), for all pairs ( k , D ) with k even or k prime and D ≢ 3 (mod 4), for all pairs with D ≤ 9 or k ≤ 4, and for a few other pairs. in the remaining cases, we obtain lower and upper bounds for n 0 (k, D).