Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecular structures, such as the Nobel prize winning fullerenes, occurring in chemistry and crystallography. The chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules. Here the authors give access to new results in the theory of polycycles and two-faced maps together with the relevant background material and mathematical tools for their study. Organised so that, after reading the introductory chapter, each chapter can be read independently from the others, the book should be accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography. Many of the results in the subject require the use of computer enumeration; the corresponding programs are available from the author's website.

• The first book on the subject of polycycles; contains all relevant background material and mathematical tools for their study • Accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography • Programs for these results are available from the web page http://www.liga.ens.fr/~dutour/BOOK_Polycycles/

### Contents

Preface; 1. Introduction; 2. Two-faced maps; 3. Fullerenes as tilings of surfaces; 4. Polycycles; 5. Polycycles with given boundary; 6. Symmetries of polycycles; 7. Elementary polycycles; 8. Applications of elementary decompositions to (r, q)-polycycles; 9. Strictly face-regular spheres and tori; 10. Parabolic weakly face-regular spheres; 11. Generalities on 3-valent face-regular maps; 12. Spheres and tori, which are aRi; 13. Frank-Kasper spheres and tori; 14. Spheres and tori, which are bR1; 15. Spheres and tori, which are bR2; 16. Spheres and tori, which are bR3; 17. Spheres and tori, which are bR4; 18. Spheres and tori, which are bRj for j ≥ 5; 19. Icosahedral fulleroids.

### Review

'… a rich source of chemical graphs (and beyond) and their properties. It should thus serve as a standard reference for researchers in the area.' Mathematical Reviews