Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurations

TL;DR

This paper explores the combinatorial properties of V-construction applied to admissible polytopes and unit-distance graphs, leading to new isometric point-circle configurations, including an infinite series related to Clifford configurations.

Contribution

It introduces the concept of admissible polytopes and analyzes their Levi graphs, revealing their relation to Kronecker covers and constructing novel isometric point-circle configurations.

Findings

Levi graph of V-construction is the Kronecker cover of the 1-skeleton.

Constructed an infinite series of point-circle configurations.

Configurations are subconfigurations of Clifford configurations.

Abstract

We call a polytope P of dimension 3 admissible if it has the following two properties: (1) for each vertex of P the set of its first-neighbours is coplanar; (2) all planes determined by the first-neighbours are distinct. It is shown that the Levi graph of a point-plane configuration obtained by V-construction from an admissible polytope P is the Kronecker cover of its 1-skeleton. We investigate the combinatorial nature of the V-construction and use it on unit-distance graphs to construct novel isometric point-circle configurations. In particular, we present an infinite series whose all members are subconfigurations of the renowned Clifford configurations.