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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.

##### MSC:
 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.)
##### Keywords:
graph spectra; automorphism group; Klein graph; Dyck graph
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