Novel highly symmetrical trivalent graphs which lead to negative curvature carbon and boron nitride chemical structures Original Research Article

A graph described by Klein in the 19th century consisting of 24 heptagons can be used to generate possible but not yet experimentally realized carbon structures through such a leapfrog transformation. The automorphism group of the Klein graph is the simple PSL(2,7) group of order 168, which can be generated from 2×2 matrices in a seven-element finite field F7 analogous to the generation of the icosahedral group of order 60 by a similar procedure using F5. Similarly, a graph described by Walther Dyck, also in the 19th century, consisting of 12 octagons on a genus 3 surface can, generate possible carbon or boron nitride structures consisting of hexagons and octagons through a leapfrog transformation. The automorphism group of the Dyck graph is a solvable group of order 96 but does not contain the octahedral group as a normal subgroup and is not a normal subgroup of the automorphism group of the four-dimensional analogue of the octahedron.