Clusters of Cycles

A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by $P(G)$, i.e. such that: (i) all interior vertices are of degree $q$, (ii) all interior faces (denote their number by $p_r$) are combinatorial $r$-gons and (implied by (i), (ii)) (iii) all vertices, edges and interior faces form a cell-complex. An example of $(r,q)$-polycycle is the skeleton of $(r^q)$, i.e. of the $q$-valent partition of the sphere $S^2$, Euclidean plane $R^2$ or hyperbolic plane $H^2$ by regular $r$-gons. Call {\it spheric} pairs $(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$; for those five pairs $P(r^q)$ is $(r^q)$ without the exterior face; otherwise $P(r^q)=(r^q)$. We give here a compact survey of results on $(r,q)$-polycycles.Comment: 21. to in appear in Journal of Geometry and Physic

THEORY OF POLYPENTAGONS

Polypentagons are systems consisting of pentagons exclusively. Some of their topological properties are studied, including the relations between certain invariants. Complete mathematical solutions are reported for the numbers of polypentagons within certain classes: catacondensed systems (without internal vertices) and systems with one internal vertex and with two connected internal vertices. A complete account on proper polypentagons is given. These systems can, by definition, be embedded on a regular dodecahedron. It is found that exactly 39 such systems exist. Their chemical formulas (C(n)H(s)), forms, and symmetries are specified