Zhang Fuji

Z -transformation graphs of perfect matchings of hexagonal systems

Abstract Let H be a hexagonal system. We define the Z -transformation graph Z( H ) to be 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 . We prove that Z ( H ) is a connected bipartite graph if H has at least one perfect matching. Furthermore, Z ( H ) is either an elementary chain or graph with girth 4; and Z ( H ) - V m is 2-connected, where V m is the set of monovalency vertices in Z ( H ). Finally, we give those hexagonal systems whose Z -transformation graphs are not 2-connected.

“Crowns”, and aromatic sextets in primitive coronoid hydrocarbons

SummaryA type of graphs derived from a cycle and associated with primitive coronoids are referred to as “crowns”. The characteristic polynomials and matching polynomials of crowns are studied. These notions are used to calculate the sextet polynomial for primitive coronoids. Patterns of aromatic sextets are treated in some detail.ZusammenfassungEine Graphentype, die von einem Cyclus abgeleitet ist und mit einfachen Coronoiden verknüpft ist, wird als “Crown” bezeichnet. Die charakteristischen Polynome und die „matching“ Polynome der Crowns werden untersucht. In diesem Rahmen werden die Sextett-Polynome für einfache Coronoide berechnet. Die Muster der aromatischen Sextette werden im Detail behandelt.

Generalized hexagonal systems with each hexagon being resonant

Abstract An edge of a generalized hexagonal system H is said to be not fixed if it belongs to some but not all perfect matchings of H . In this paper we give a necessary and sufficient condition for a generalized hexagonal system in which every edge is not fixed. Applying the above result to complete generalized hexagonal systems, we obtain a simple criterion to determine whether or not each hexagon of a complete generalized hexagonal system is resonant, and give a new and simpler proof of the main theorem of [4].Zhang, F. and M. Zheng, Generalized hexagonal systems with each hexagon being resonant, Discrete Applied Mathematics 36 (1992) 67-73. An edge of a generalized hexagonal system H is said to be not fixed if it belongs to some but not all perfect matchings of I-I. In this paper we give a necessary and sufficient condition for a generalized hexagonal system in which every edge is not fixed. Applying the above result to complete generalized hexagonal systems, we obtain a simple criterion to determine whether or not each hexagon of a complete generalized hexagonal system is resonant, and give a new and simpler proof of the main theorem of [4]. Since the hexagonal system with at least one perfect matching can be regarded as the skeleton of a benzenoid hydrocarbon molecule, the hexagonal system has been studied by mathematicians and chemists. The reader interested in thz subject may consult [1,3]. From resonance theory and Clar’s theory of the aromatic sextet, in general not every hexagon of a hexagonal system is resonant. in j4], tk hexagonal systems for which each hexagon is resonant are characterized. This paper considers the case of generalized hexagonal systems, i.e., the hexagonal systems having “holes”. A hexagonal system (HS) is a finite 2-connected plane graph in which each interior face is a regular hexagon of side length 1. All hexagons which appear in this paper are regular and have side length 1. Undefined terminology can be found in [4]. A matching M of a graph G is a set of indepeedent edges of G. If each vertex of G is incident with an edge of M, then M is called a perfect matching. A generalized hexagonal system (GHS) is a graph obtained by deleting some interier vertices and interior edges from a hexagonal system. All generalized hexagonal systems considered in this paper have perfect matchings. * Present address: RUTCOR, Rutgers University, P.O. Box 5062, New Brunswick, NJ 08903, USA. 0166-218X/92/

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

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Perfect matchings in 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).

Directed tree structure of the set of Kekule´ patterns of generalized polyhex graphs

This paper deals with perfect matchings in hexagonal systems. Counterexamples are given to Sachss conjecture in this field. A necessary and sufficient condition for a hexagonal system to have a perfect matching is obtained.

Theory of helicenic hydrocarbons. Part 1: Invariants and symmetry

Zhang, F. and X. Guo, Directed tree structure of the set of Kekule patterns of generalized polyhex graphs, Discrete Applied Mathematics 32 (1991) 295-302. In this paper we define the concept of g-sextet rotation graph of a generalized polyhex graph G, and prove that the g-sextet rotation graph D(G) of G is a directed tree. This conclusion is a generalization of [l] and also valid for any polyhex fragment graphs. Furthermore, by our results, an error in a proof [2] of the Ohkami-Hosoya conjecture [3] about a one-to-one correspondence between Kekule and sextet patterns of a polyhex graph is corrected. A polyhex graph, also called hexagonal system, benzenoid system, honeycomb system, is a finite connected plane graph with no cut-vertices in which every interior region is a hexagonal unit cell [5] (a regular hexagon). In [l], a generalized polyhex graph was defined to be a graph obtained from a polyhex graph G by deleting all the vertices and edges which lie in the interiors of a group of separated cycles inside G (that is, each of separated cycles contains no vertex on the boundary of G). In other words, the set of generalized polyhex graphs contains not only the polyhex graphs but also those changed polyhex graphs with holes. In the present paper we further extend the concept of a generalized polyhex graph to a finite connected plane graph with no cut-vertices on the regular hexagonal lattice (see Fig. 1). * Supported by NNSFC. 0166-218X/91/

GRAPH-THEORETICAL STUDIES ON FLUORANTHENOIDS AND FLUORENOIDS - ENUMERATION OF SOME CATACONDENSED SYSTEMS

03.50 G 1991 Elsevier Science Publishers B.V. (North-Holland) 296 F. Zhang, X. GuoAbstract In this paper we define the concept of g -sextet rotation graph of a generalized polyhex graph G , and prove that the g -sextet rotation graph D ( G ) of G is a directed tree. This conclusion is a generalization of [1] and also valid for any polyhex fragment graphs. Furthermore, by our results, an error in a proof [2] of the Ohkami—Hosoya conjecture [3] about a one-to-one correspondence between Kekule and sextet patterns of a polyhex graph is corrected.

Recognizing Kekuléan benzenoid systems byC-P-V path elimination

Helicenes form a subclass of polyhexes and correspond to hydrocarbons of considerable chemical interest. This paper is the first part of a general graph-theoretical treatment of helicenes. The invariants are studied: the relations between them, their possible values, and their upper and lower bounds in helicenes. Extremal helicenes and circular helicenes are useful definitions of subclasses of the systems under consideration. Finally an account of symmetry of helicenes is given.

A theorem concerning perfect matchings in hexagonal systems

Abstract Precise definitions are given for some classes of molecular graphs with one pentagon and otherwise hexagons: the monopentapolyhexes. The fluoranthenoid and fluorenoid systems belong to monopentapolyhexes. Complete mathematical solutions, using combinatorial summations on the one hand and generating functions on the other hand, are given for the numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids and the corresponding helicenic systems). Generating functions and numerical values are included.