By Michel Deza, Mathieu Dutour Sikirić

Polycycles and symmetric polyhedra seem as generalizations of graphs within the modeling of molecular buildings, comparable to the Nobel prize profitable fullerenes, happening in chemistry and crystallography. The chemistry has encouraged and knowledgeable many fascinating questions in arithmetic and desktop technological know-how, which in flip have advised instructions for synthesis of molecules. the following the authors provide entry to new ends up in the speculation of polycycles and two-faced maps including the proper heritage fabric and mathematical instruments for his or her examine. geared up in order that, after interpreting the introductory bankruptcy, each one bankruptcy could be learn independently from the others, the ebook may be available to researchers and scholars in graph concept, discrete geometry, and combinatorics, in addition to to these in additional utilized components comparable to mathematical chemistry and crystallography. a number of the leads to the topic require using desktop enumeration; the corresponding courses can be found from the author's site.

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Thus, the Petersen graph is the smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Ci , Cmh , (m even), Dmh (m even), Dmd (m odd), Th , Oh , and Ih . A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list: Ci , C2h , D2h , D6h , D3d , D5d , Th , and Ih .

Spherical and toroidal fullerenes have an extensive chemical literature, and Klein bottle polyhexes have been considered, for example, in [Kir97, KlZh97]. Note that at least one spherical fullerene with v vertices exists for all even v with v ≥ 20, except for the case v = 22 ([GrMo63]). 2 Toroidal and Klein bottle fullerenes The (6, 3)-tori and (6, 3)-Klein bottles are related to {6, 3} in a straightforward way. The underlying surfaces are quotients of the Euclidean plane R2 under groups of isometries generated by two translations (for T2 ) or one translation and one glide reflection (for K2 ).

So, F and F intersect in an edge, vertex, or ∅. The same proof works for other intersections of cells. 3 If a finite (r, q)-polycycle has a boundary vertex whose degree is less than q, then the total number of these vertices is at least two. Proof. Take a flag f in this (r, q)-polycycle P and a flag f in {r, q}. 1. The boundary B of P consists of vertices v1 , . . , vk . The image of this boundary in {r, q} is also a cycle φ(B). 50 Chemical Graphs, Polycycles, and Two-faced Maps Whenever B has a vertex vi of degree q, the edges ei = vi−1 vi and ei+1 = vi vi+1 belong to a common face F.