Resolving Sets and Split Resolving Sets of Symmetric Designs

Abstract

We study resolving sets and split resolving sets of the point-block incidence graphs of symmetric designs and we obtain general lower bounds on their cardinality. In some cases, this lower bound is just a constant factor away from the known upper bounds. In particular, we show that for any $\varepsilon>0$ there exists $q_0$ and $n_0$ such that if $q\geq q_0$ and $n\geq n_0$, then the metric dimension of the point-hyperplane incidence graph of $\mathrm{PG}(n,q)$ is at least $(2-\varepsilon)nq$. The best known upper bound for the metric dimension of $\mathrm{PG}(n,q)$ is roughly $4nq$. We also prove that the metric dimension of a symmetric $(v,k,\lambda)$ design, under certain conditions, is at least $\frac{(2-\varepsilon)uv}k$ for any $\varepsilon >0$, where $u=\left\lfloor\frac{\ln v}{\ln v - \ln k +1}\right\rfloor$.