On the metric dimension of Grassmann graphs

Abstract

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and ComplexityThe metric dimension of a graph Gamma 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 G(q)(n, k) (whose vertices are the k-subspaces of F-q(n), and are adjacent if they intersect in a (k 1)-subspace) for k \textgreater= 2. We find an upper bound on its metric dimension, which is equal to the number of 1-dimensional subspaces of F-q(n). 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.