Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer–Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer–Doyen parameters.

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Let v > k > i be non-negative integers. The generalized Johnson graph, J(v, k, i), is the graph whose vertices are the k-subsets of a v-set, where vertices A and B are adjacent whenever |A ∩ B| = i. In this article, we derive general formulas for the girth and diameter of J(v, k, i). Additionally, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection.

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