Metric dimension of dual polar graphs

TL;DR

This paper investigates the metric dimension of dual polar graphs, establishing an upper bound based on the rank of their incidence matrices and providing explicit calculations for these bounds.

Contribution

It introduces a novel upper bound for the metric dimension of dual polar graphs using incidence matrix ranks and computes these bounds explicitly.

Findings

Upper bound on metric dimension via incidence matrix rank

Explicit calculations for dual polar and halved dual polar graphs

Connection between graph metric dimension and algebraic properties of polar spaces

Abstract

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\mu(\Gamma)$ is the smallest size of a resolving set for $\Gamma$. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over $\mathbb{R}$ of the incidence matrix of the corresponding polar space. We then compute this rank to give an explicit upper bound on the metric dimension of dual polar graphs, as well as the halved dual polar graphs.