Chen Rongsi

The connectivity ofZ-transformation graphs of perfect matchings of hexagonal systems

LetH be a hexagonal system. TheZ-transformation graphZ(H) is a graph where the vertices are perfect matchings ofH and where two perfect matchings are joined by an edge provided their synimetric difference consists of six edges of a hexagon ofH. We prove that the connectivity ofZ(H) is equal to the minimum degree of vertices ofZ(H).

Hamilton paths in Z -transformation graphs of perfect matchings of hexagonal systems

Abstract Let H be a hexagonal system. The Z-transformation graph Z(H) is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H (Z. Fu-ji et al., 1988). In this paper we prove that Z(H) has a Hamilton path if H is a catacondensed hexagonal system. A hexagonal system [ll], also called honeycomb system or hexanimal (see, eg. [lo]) is a finite connected plane graph with no cut-vertices, in which every interior region is surrounded by a regular hexagon of side length 1. Hexagonal systems are of chemical significance since a hexagonal system with perfect matchings is the skeleton of a benzenoid hydrocarbon molecule [9]. Recall that a perfect matching of a graph G is a set of disjoint edges of G covering all the vertices of G. In the following discussion we confine our considerations to those hexagonal systems with at least one perfect matching. Let H be a hexagonal system. The Z-transformation graph Z(H) [3,4] is the graph where the vertices are the perfect matchings of H and where two perfect matchings

Methods of enumerating Kekulé structures, exemplified by applications to rectangle-shaped benzenoids

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the “perforated” oblate rectangles, is considered in order to exemplify a “perfectly explicit” combinatorial K formula, an expression for arbitraty values of the parameters m and n.

Recognition of essentially disconnected benzenoids

A benzenoid, or a benzenoid system, is a connected planar graph whose every interiorface is a regular hexagon. A peak (resp. valley) of a benzenoid is a vertex which lies above(resp. below) all its first neighbors. A Kekulean benzenoid is a benzenoid with at least oneperfect matching. An essentially disconnected benzenoid is a Kekulean benzenoid which hassome fixed bonds. Essentially disconnected benzenoids have proved to be very useful incertain enumeration techniques for Derfect matching. Hence the problem of recognizing

Regular coronoid systems

Abstract A coronoid system H can be regarded as a hexagonal system that is allowed to have ‘holes’ such that the perimeter of H and the perimeters of the holes are pairwise disjoint. H is said to be normal if it has no fixed bond. A normal coronoid system is called regular if it can be constructed from a smaller one that has already been recognized as regular by adding a hexagon in some special ways. In this paper a necessary and sufficient condition for a coronoid system to be regular is given.

A COMPLETE SOLUTION OF HOSOYA'S MYSTERY

In this note a theorem concerning the coincidence between the characteristic polynomial of a cycle and the polynomial of Kekulé structure count of a primitive coronoid is presented which implies a complete solution of Hosoyas mystery.

Fixed Bonds in Indacenoids

Abstract Indacenoids are polygonal systems consisting of two pentagons each and arbitrary numbers of hexagons. They represent a class of polycyclic conjugated hydrocarbons. A fixed bond of a polygonal system is an edge which is or is not selected in all the Kekule structures of that polygonal system. It is proved that each catacondensed indacenoid has some fixed single bonds, but has no fixed double bonds. Moreover, the location of fixed single bonds in a catacondensed indacenoid is identified. Finally, a theorem about fixed single bonds in pericondensed indacenoids is proved.

Recognizing essentially disconnected benzenoids with fixed double bonds

A Kekuléan benzenoid system is one with Kekulé structures. A fixed double (single) bond of a Kekuléan benzenoid systemH is an edge belonging to all (none) of the Kekulé structures ofH. Essentially disconnected systems are Kekuléan pericondensed benzenoid systems with some fixed double or single bonds. In this paper a necessary and sufficient condition for a Kekuléan benzenoid system to be essentially disconnected benzenoid system with fixed double bonds is given and rigorously proved.

Enumeration of kekulé structures: perforated rectangles

Abstract The Kekule structures of multiple coronoids were studied. The main class consists of oblate rectangles R j ( m,n ) with ( m −1) holes of the size of ( n −2) hexagons each. A general combinatorial formula for the Kekule structure counts ( K ) is deduced. Studies of K numbers for some related classes are included.

Benzenoids with branching graphs being trees

Abstract The branching graph of a benzenoid hydrocarbon molecule is the subgraph consisting of vertices of degree 3 and the bonds among them. In this paper, a practical method is established for recognizing trees embedded in the infinite hexagonal lattice which can be branching graphs of benzenoids. Moreover, a simple criterion is given to determine when a tree embedded in the infinite hexagonal lattice is the branching graph of a catacondensed benzenoid.