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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Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over F q . Let G := V ⋊ G 0 , where G 0 is an irreducible subgroup of GL (V ) which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs Γ that admit such a group G as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G 0 is a subgroup of either ΓL (n, q) or ΓSp (n, q) and is maximal in one of the Aschbacher classes C i , where i ∈ {2, 4, 5, 6, 7, 8}. We are able to determine all graphs Γ which arise from G 0 ≤ ΓL (n, q) with i ∈ {2, 4, 8}, and from G 0 ≤ ΓSp (n, q) with i ∈ {2, 8}. For the remaining classes we give necessary conditions in order for Γ to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs.

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We investigate locally n×n grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2n by previous work of Blokhuis and Brouwer. We show that if each distance two pair is joined by at least n − 1 paths of length 2 then the diameter is bounded by O(log(n)), while if each pair is joined by at least 2(n − 1) such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally n × n grid for odd prime powers n, and apply these results to locally 5 × 5 grid graphs to obtain a classification for the case where either all µ-graphs have order at least 8 or all µ-graphs have order c for some constant c.

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