Geometric Structure of Chemistry-relevant Graphs: zigzags and central circuits

Springer 2 This book is a companion to our book [DeDu08], which considered the notions of polycycles, face-regularity and weak face-regularity of plane graphs and toroidal maps. The central actors in the present monograph are the zigzags and central-circuits of 3-or 4-regular plane graphs, which allow to obtain a double covering or covering of the edge-set. This study is mainly focused on specific classes of bifaced plane graphs, i.e. those without faces of negative curvature. This contains, as a particular case, the fullerenes, i.e., 3-regular plane graphs with faces of size 5 or 6, which are prominent Chemistry-relevant graphs. The class also contains the octahedrites, i.e., 4-regular plane graphs with faces of size 3 or 4. We also consider three classes of graphs which are self-dual. For those graphs we consider how to enumerate them, their possible symmetry groups, their connectivity and other structural properties. We also study the icosahedrites, i.e. 5-regular plane graphs with faces of size 3 or 4; those have faces of negative curvature and so, their number grows exponentially. Finally, we consider disk-fullerenes, i.e. 3-regular partitions of a disk by 5-and 6-gons. For all those classes of graphs, we treat the notion of zigzags and central-circuits; sometimes, at the same time. We consider simplicity of circuits, possible configuration , tightness and enumeration of the tight graphs with simple circuits. We also address extremal questions, such as the maximum number of circuits of tight graphs. For the classes of graphs with maximal symmetry, such as the fullerenes of icosa-hedral symmetry, a special construction, named Goldberg-Coxeter construction, allows to describe them explicitly in term of two integer parameters k and l. This construction is studied systematically for 3-, 4-and 6-valent graphs and allow us to describe many classes in a simple way. We study the zigzags and central-circuits of the obtained graphs and build a new (k, l)-product algebraic formalism that allow us to describe the zigzags and central-circuits of the obtained graphs explicitly. For classes of graphs with non-maximal symmetry more complex description are needed. We explain how this can be done in practice by presenting the formalism of hyperbolic complex geometry derived by William Thurston in [Thur98]. For dimensions higher than 2, the possible similar structures are more complicated. In that case, we limit ourselves to zigzags and compute them for several infinite families of complexes and the regular, semiregular and regular-faced polytopes.