Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurations - Gévay, Gábor (Ars mathematica contemporanea, 2014)
We call a convex 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 the 1-skeleton of ?$P$?. 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 all of whose members are subconfigurations of the renowned Clifford configurations.
Keywords: Clifford configuration; Cliffordova konfiguracija; Danzer configuration; Danzerjeva konfiguracija; generalized cuboctahedron graph; grafi z enotsko razdaljo; isometric point-circle configuration; Kronecker cover; Kroneckerjev krov; unit-distance graph; V-construction; V-konstrukcija