Isometric point-circle configurations on surfaces from uniform maps

TL;DR

This paper explores the embedding of neighborhood geometries of graphs on surfaces as point-circle configurations, providing new examples from regular maps, surveys existing geometries, and presenting an infinite family of configurations with specific symmetries.

Contribution

It introduces new point-circle configurations derived from regular maps and describes an infinite family with particular automorphism properties.

Findings

Examples from regular maps with maximum automorphisms

Survey of pentagonal geometries from Moore graphs

Infinite family of configurations on p-gonal surfaces

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.