On the metric dimension of Grassmann graphs

TL;DR

This paper establishes an upper bound on the metric dimension of Grassmann graphs, showing it equals the number of 1-dimensional subspaces of the underlying vector space, which aids in understanding their structural properties.

Contribution

The paper provides a constructive upper bound on the metric dimension of Grassmann graphs, linking it to the count of 1-dimensional subspaces in the vector space.

Findings

Upper bound on metric dimension equals the number of 1-dimensional subspaces

Constructive method for determining the bound

Enhances understanding of Grassmann graph structure

Abstract

The {\em 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 $\mathbb{F}_q^n$, and are adjacent if they intersect in a $(k-1)$-subspace) for $k\geq 2$, and find a constructive upper bound on its metric dimension. Our bound is equal to the number of 1-dimensional subspaces of $\mathbb{F}_q^n$.