Isometric point-circle configurations on surfaces from uniform maps

Abstract

We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with maximum number of automorphisms for their genus and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle $v_4$ configurations on $p$-gonal surfaces with two $p$-gonal morphisms is given. The image of these configuration on the sphere under the two $p$-gonal morphisms is also described.