Adjacency preserving mappings on point-line geometries

Abstract

In certain point-line geometries $G=\left(\mathcal{P},\mathcal{ℒ}\right)$, $G^{\prime }=\left({\mathcal{P}}^{\prime },{\mathcal{ℒ}}^{\prime }\right)$, which include dense near polygons and the geometry of Hermitian matrices, any adjacency preserving mapping $\phi :\mathcal{P}\to {\mathcal{P}}^{\prime }$ satisfying

$$\exists p,q\in \mathcal{P}:d\left(p,q\right)=d\left(p^{\phi },q^{\phi }\right)=diam\left(G\right),$$

is an isomorphism between $G$ and $\phi \left(G\right)$.