Geometric structure of chemistry-relevant graphs. Zigzags and central circuits. (English) Zbl 1320.05002

Deza, Michel-Marie; Dutour Sikirić, Mathieu; Shtogrin, Mikhail Ivanovitch

Forum for Interdisciplinary Mathematics 1. New Delhi: Springer (ISBN 978-81-322-2448-8/hbk; 978-81-322-2449-5/ebook). xi, 211 p. (2015).

The present book focuses on zigzags and central circuits of three or four-regular plane graphs, which allow to obtain a double covering or covering of the edgeset. Mainly, specific classes of bifaced plane graphs (i.e., those without faces of negative curvature) that are of great importance in mathematical chemistry are studied. It contains, for example, fullerenes, octahedrites, icosahedrites and disk-fullerenes. For these classes of graphs, the zigzags and central circuits are considered. Furthermore, a special construction called Goldberg-Coxeter construction is studied systematically for three-, four-, and six-valent graphs. It allows to describe them explicitly in terms of two integer parameters \(k\) and \(l\). For classes of graphs with non-maximal symmetry, more complex description is explained in practise by presenting the formalism of hyperbolic complex geometry. Moreover, for several infinite families of complexes and polytopes the zigzag structure is computed.

Reviewer: Niko Tratnik (Maribor)

MSC:

05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
05C90	Applications of graph theory
92E10	Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
94C05	Analytic circuit theory

Keywords:

zigzags; central circuits; bifaced plane graphs; fullerenes; Goldberg-Coxeter construction; complexes; polytopes