We consider the representation of a continuous-time quantum walk in a graph X by the matrix exp(−itA(X)). We provide necessary and sufficient criteria for distance-regular graphs and, more generally, for graphs in association schemes to have perfect state transfer. Using these conditions, we provide several new examples of perfect state transfer in simple graphs.

We consider Erdős-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect nontrivially. We characterize the Erdős-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n + 1, q 2 ) with n 2.

In this paper we investigate partial spreads of H(2n−1, q 2 ) through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of H(2n − 1, q 2 ). We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of H(3, q 2 ) for a range of sizes.2010 Mathematics Subject Classification. 05B25, 51E23, 51A50, 15A03.

The inequality of Higman for generalized quadrangles of order (s, t) with s > 1 states that t ≤ s 2 . We generalize this by proving that the intersection number c i of a regular near 2d-gon of order (s, t) with s > 1 satisfies the tight bound c i ≤ (s 2i − 1)/(s 2 − 1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d, b, α, β) = (d, −q, −(q + 1)/2, −((−q) d + 1)/2) with q an odd prime power.We refer the reader to Sect. 2 for the definitions of, for instance, (finite) generalized polygons, near polygons, and polar spaces.Feit and Higman [18] showed that (finite) generalized n-gons of order (s, t) = (1, 1) with n ≥ 3 can only exist if n ∈ {3, 4, 6, 8, 12}; if n = 12, then s = 1 or t = 1. If s > 1, then the following inequalities must hold: if n = 4, then t ≤ s 2 [20], if n = 6, then t ≤ s 3 [19], and if n = 8, then t ≤ s 2 [21]. Bose and Shrikhande [5] also proved that if n = 4 and t = s 2 , then for any triple of nonadjacent vertices, the number

We prove that in every finite Hermitian polar space of odd dimension and even maximal dimension $\rho$ of the totally isotropic subspaces, a partial spread has size at most $q^{\rho+1}+1$, where $GF(q^2)$ is the defining field. This bound is tight and is a generalisation of the result of De Beule and Metsch for the case $\rho=2$.