2012/05/01 by Arjana Žitnik, Boris Horvat, Tomaž Pisanski · 2 citations
Computer Science · Mathematics · Engineering · #Advanced Graph Theory Research #Finite Group Theory Research #graph theory and CDMA systems #Combinatorics #Mathematics #Euclidean geometry #Symmetric graph #Distance-regular graph #Graph #Discrete mathematics #Voltage graph #Cubic graph #Line graph #Geometry
paper · pdf · doi:10.4134/jkms.2012.49.3.475
openalex publication_date 2012/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of <TEX>I</TEX>-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each <TEX>I</TEX>-graph <TEX>I(n,j,k)</TEX> admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every <TEX>I</TEX>-graph <TEX>I(n,j,k)</TEX> has an isomorphic <TEX>I</TEX>-graph that admits a unit-distance representation in the Euclidean plane with a <TEX>n</TEX>-fold rotational symmetry, with the exception of the families <TEX>I(n,j,j)</TEX> and <TEX>I(12m,m,5m)</TEX>, <TEX>m≥1</TEX>. We also provide unit-distance representations for these graphs.