Unbranched Catacondensed Polygonal Systems Containing Hexagons and Tetragons

An algebraic solution for the isomer numbers of unbranched a-4- catafusenes is presented. An a-4-catafusene is a catacondensed polygonal system consisting of exactly o: tetragons each and otherwise only hexagons. This analysis, which makes use of certain triangular matrices including the Pascal triangle, is a continuation of a previous work on di-4-catafusenes. By serendipity, the problem was reversed in the sence that the systems were considered as possessing \u277 hexagons each and otherwise only tetragons. Under this viewpoint the enumeration problem could be solved more directly and led to explicit formulas. Finally, the resuIts are applied to catafusenes as a special case

A New Class of Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons

Di-4-catafusenes are defined as catacondensed polygonal systems consisting of two tetragons each and otherwise only hexagons. Di- 4-catafusenes are enumerated by combinatorial constructions and by computer programming. For the unbranched systems (nonhelicenic + helicenic), as the main result of the present work, a complete mathematical solution is reported. A new algebraic approach has been employed, which involves a triangular matrix with some interesting mathematical properties

Unbranched Catacondensed Polygonal Systems Containing Hexagons and Tetragons

An algebraic solution for the isomer numbers of unbranched a-4- catafusenes is presented. An a-4-catafusene is a catacondensed polygonal system consisting of exactly o: tetragons each and otherwise only hexagons. This analysis, which makes use of certain triangular matrices including the Pascal triangle, is a continuation of a previous work on di-4-catafusenes. By serendipity, the problem was reversed in the sence that the systems were considered as possessing \u277 hexagons each and otherwise only tetragons. Under this viewpoint the enumeration problem could be solved more directly and led to explicit formulas. Finally, the resuIts are applied to catafusenes as a special case

Isomers of Polyenes Attached to Benzene

A polyene graph is a tree that can be embedded in a hexagonal lattice. Systems of polyene graphs attached to one hexagon are considered. Overlapping edges and/or vertices (geometrically nonplanar systems) are allowed. A complete mathematical solution is presented in terms of a generating function for the numbers of isomers of the systems in question. The corresponding geometrically planar systems, referred to as styrenoids, are enumerated by computer programming. Finally, in the Appendix, the generating function is given for the numbers of free polyene graphs

Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

Enumeration and Classification of Double Coronoid Hydrocarbons Appendix: Triple Coronoids

An enumeration of double coronoids (polyhexes with two heles) is performed, both by hand and by computer. The numbers 15123 and 125760 for the systems with h = 17 and h = 18, respectively, are reported for the first time. Here h denotes the number of hexagons. The generated systems are classified in different ways. In this connection the strata of corona hale constellations are defined, depending on the proximity of the holes. As appendix, a first enumeration of triple coronoids is reported

All-Benzenoid Systems: an Algebra of Invariants

The current invariants of all-benzenoids (h, n, m, nu na s - defined in the Introduction), in addition to the number of full and of empty hexagons (v and t), are studied. Their possible values are specified. Some of the relations between these invariants are summarized in a systematic way. The upper and lower bounds for all of them are accounted for as functions of any of these invariants

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