Book contents
- Frontmatter
- 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 General properties of 3-valent face-regular maps
- 12 Spheres and tori that are aRi
- 13 Frank-Kasper spheres and tori
- 14 Spheres and tori that are bR1
- 15 Spheres and tori that are bR2
- 16 Spheres and tori that are bR3
- 17 Spheres and tori that are bR4
- 18 Spheres and tori that are bRj for j ≥ 5
- 19 Icosahedral fulleroids
- References
- Index
4 - Polycycles
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- 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 General properties of 3-valent face-regular maps
- 12 Spheres and tori that are aRi
- 13 Frank-Kasper spheres and tori
- 14 Spheres and tori that are bR1
- 15 Spheres and tori that are bR2
- 16 Spheres and tori that are bR3
- 17 Spheres and tori that are bR4
- 18 Spheres and tori that are bRj for j ≥ 5
- 19 Icosahedral fulleroids
- References
- Index
Summary
(r, q)-polycycles
A (r, q)-polycycle is a simple plane 2-connected locally finite graph with degree at most q, such that:
(i) all interior vertices are of degree q,
(ii) all interior faces are (combinatorial) r-gons.
We recall that any finite plane graph has a unique exterior face; an infinite plane graph can have any number of exterior faces, including zero and infinity. Denote by pr the number of interior faces; for example, Dodecahedron on the plane has p5 = 11.
See in Figure 4.1 some examples of connected simple plane graphs that are not (r, q)-polycycles.
We will prove later (in Theorem 4.3.2) that all vertices, edges, and interior faces of an (r, q)-polycycle form a cell-complex (see Section 1.2.1).
The skeleton of a polycycle is the edge-vertex graph defined by it, i.e. we forget the faces. By Theorem 4.3.6, except for five Platonic ones, the skeleton has a unique polycyclic realization, i.e. a polycycle for which it is the skeleton.
The parameters (r, q) are called elliptic if rq < 2(r + q), parabolic if rq = 2(r +q), and hyperbolic if rq > 2(r +q); see Remark 1.4.1. Call a polycycle outerplanar if it has no interior vertices. For parabolic or hyperbolic (r, q), the tiling {r, q} is a (r, q)-polycycle. For elliptic (r, q), the tiling {r, q} with a face deleted is an (r, q)- polycycle. Different, but all isomorphic, polycyclic realizations for those five exceptions to the unicity, come from different choices of such deleted (exterior) faces.
- Type
- Chapter
- Information
- Geometry of Chemical GraphsPolycycles and Two-faced Maps, pp. 43 - 55Publisher: Cambridge University PressPrint publication year: 2008