On the metric dimension of Grassmann graphs

The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph Gq(n,k) (whose vertices are the k-subspaces of Fq, and are adjacent if they intersect in a (k− 1)-subspace) for k ≥ 2. We find an upper bound on its metric dimension, which is equal to the number of 1dimensional subspaces of Fq. We also give a construction of a resolving set of this size in the case where k +1 divides n, and a related construction in other cases.