Configuration Graphs of Neighbourhood Geometries

Congurations of type $(\kappa^2 +1)_{\kappa}$ give rise to $\kappa$--regular simple graphs via conguration graphs. On the other hand, neighbourhood geometries of $C_4$--free $\kappa$--regular simple graphs on $\kappa^2 +1$ vertices turn out to be congurations of type $(\kappa^2 +1)_{\kappa}$. We investigate which congurations of type $(\kappa^2 +1)_{\kappa}$ are equal or isomorphic to the neighbourhood geometry of their conguration graph and conversely. We classify all such graphs and congurations for $\kappa = 3$ and for $\kappa = 4$ when the graphs admit a centre of radius 2.