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.

### Configuration Graphs of Neighbourhood Geometries

#### Abstract

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.
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2008
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/9231
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