On the metric dimension of Grassmann graphs

Robert F Bailey,Karen Meagher

doi:10.46298/dmtcs.532

Abstract

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity The 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.

Highlights

We are concerned with finding an upper bound on the metric dimension of Grassmann graphs
Definition 4 The Grassmann graph Gq(n, k) has as its vertex set the set of all k-dimensional subspaces of V (n, q), and two vertices are adjacent if the corresponding subspaces intersect in a subspace of dimension k − 1
The disadvantage of the proof of Theorem 5 given above is that it does not provide an explicit construction of a resolving set for the Grassmann graph Gq(n, k)

Summary

Introduction

We are concerned with finding an upper bound on the metric dimension of Grassmann graphs. Definition 2 The metric dimension of Γ, denoted μ(Γ), is the smallest size of a resolving set for Γ. In the case of primitive distance-regular graphs, bounds on a parameter equivalent to the metric dimension were obtained in 1981 by Babai [1] Many families of distance-regular graphs are so-called “graphs with classical parameters” (see [5, Chapter 9]): these include the well-known Hamming graphs and Johnson graphs, for which metric dimension has already been studied. Throughout this paper, V (n, q) denotes the n-dimensional vector space over the finite field Fq. Definition 4 The Grassmann graph Gq(n, k) has as its vertex set the set of all k-dimensional subspaces of V (n, q), and two vertices are adjacent if the corresponding subspaces intersect in a subspace of dimension k − 1. This bound, and its proof, are inspired by a similar result for the Johnson graph J(n, k) given in Theorem 3 above

Proof of the bound

Constructions of resolving sets

Discussion