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Given a finite Lie incidence geometry which is either a polar space of rankat least $3$ or a strong parapolar space of symplectic rank at least $4$ anddiameter at most $4$, or the parapolar space arising from the line Grassmannianof a projective space of dimension at least $4$, we show that its point graphis determined by its local structure. This follows from a more general resultwhich classifies graphs whose local structure can vary over all localstructures of the point graphs of the aforementioned geometries. In particular,this characterises the strongly regular graphs arising from the lineGrassmannian of a finite projective space, from the half spin geometry relatedto the quadric $Q^+(10,q)$ and from the exceptional group of type$\mathsf{E_6}(q)$ by their local structure.

Local recognition of the point graphs of some Lie incidence geometries