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Novel highly symmetrical trivalent graphs which lead to negative curvature carbon and boron nitride chemical structures. (English) Zbl 0994.05144
Summary: 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 \(\text{PSL}(2,7)\) group of order 168, which can be generated from \(2\times 2\) matrices in a seven-element finite field \({\mathcal F}_7\) analogous to the generation of the icosahedral group of order 60 by a similar procedure using \({\mathcal F}_5\). 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.

05C90 Applications of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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