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### Z-Transformation Graphs of Perfect Matchings of Hexagonal Systems [chapter]

Zhang Fu-Ji, Guo Xiao-Feng, Chen Rong-Si
1988 Annals of Discrete Mathematics
In this paper we introduce a new kind of transformation graph, namely Z-transformation graph of perfect matchings of hexagonal systems and observe some of its properties.  ...  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  ...

### Z-transformation graphs of perfect matchings of hexagonal systems

Zhang Fu-ji, Guo Xiao-feng, Chen Rong-si
1988 Discrete Mathematics
In this paper we introduce a new kind of transformation graph, namely Z-transformation graph of perfect matchings of hexagonal systems and observe some of its properties.  ...  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  ...

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

Rong-si Chen, Fu-ji Zhang
1997 Discrete Applied Mathematics
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  ...  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 Ml and M2 are joined by an edge provided their symmetric difference  ...

### Plane elementary bipartite graphs

Heping Zhang, Fuji Zhang
2000 Discrete Applied Mathematics
Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous  ...  of G) and whose Z-transformation graphs Z(G) contain vertices of degree one. ?  ...  In connection to resonant hexagons (or aromatic sextets), Z-transformation graphs of perfect matchings of hexagonal systems H were deÿned [27, 28] .  ...

### Forcing faces in plane bipartite graphs

Zhongyuan Che, Zhibo Chen
2008 Discrete Mathematics
[Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl.  ...  Using the tool of Z-transformation graphs developed by Zhang et al.  ...  For a hexagonal system H, we proved in  that if H has a perfect matching M such that M is of degree two in Z(H ) and the two M-resonant hexagons are adjacent, then Z(H ) is a path.  ...

### Page 1318 of Mathematical Reviews Vol. , Issue 90B [page]

1990 Mathematical Reviews
The paper under review defines the Z- transformation graph Z(H) of a hexagonal system H with at least one perfect matching to be the graph whose vertices are the perfect matchings of H such that two perfect  ...  Trinajstié (Zagreb) 90b:92083 92A40 05C70 Zhang, Fu Ji (PRC-XINJ); Guo, Xiao Feng (PRC-XINJ); Chen, Rong Si (PRC-FZHU-FE) Z-transformation graphs of perfect matchings of hexagonal systems.  ...

### Z -transformation graphs of maximum matchings of plane bipartite graphs

Heping Zhang, Rijun Zha, Haiyuan Yao
2004 Discrete Applied Mathematics
If G has a perfect matching, Z(G) andZ(G) are the usual Z-transformation graph and digraph. If G has neither isolated vertices nor perfect matching, then Z(G) is not connected.  ...  Let G be a plane bipartite graph. The Z-transformation graph Z(G) and its orientationZ(G) on the maximum matchings of G are deÿned.  ...  [6, 7] introduced a concept of Z-transformation graph on the set of perfect matchings of a hexagonal system: two vertices are adjacent provided that their corresponding perfect matchings only di er  ...

### Forcing faces in plane bipartite graphs (II)

Zhongyuan Che, Zhibo Chen
2013 Discrete Applied Mathematics
Che and Z. Chen, Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (2006) 649-668].  ...  We also give a new necessary and sufficient condition for a finite face of G to be forcing in terms of bridges in the Z -transformation graph Z (G) of G.  ...  In 1995, Zhang and Li  gave characterizations for a hexagonal system with forcing edges, by using the concept of Z -transformation graph of a hexagonal system introduced by Zhang et al. in   ...

### Z-transformation graphs of perfect matchings of plane bipartite graphs

Heping Zhang, Fuji Zhang, Haiyuan Yao
2004 Discrete Mathematics
The Z-transformation graph, ZF (G), of G with respect to a speciÿc set F of faces is deÿned as a graph on the perfect matchings of G such that two perfect matchings M1 and M2 are adjacent provided M1 and  ...  Let G be a plane bipartite graph with at least two perfect matchings.  ...  [11, 12] introduced the concept of Z-transformation graphs, Z(H ), of perfect matchings of hexagonal systems H and showed that the connectivity of Z(H ) is equal to the minimum degree.  ...

### Page 6746 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews
(PRC-LAN; Lanzhou Z-transformation graphs of maximum matchings of plane bipartite graphs.  ...  The Z-transformation graph Z(G) of a plane graph G is defined as the simple graph whose vertices are the maximum matchings of G, two vertices being adjacent if and only if their symmetric difference is  ...

### A relation between Clar covering polynomial and cube polynomial [article]

Heping Zhang, Wai-Chee Shiu, Pak-Kiu Sun
2012 arXiv   pre-print
In this paper we find that the Clar covering polynomial of a hexagonal system H coincides with the cube polynomial of its resonance graph R(H) by establishing a one-to-one correspondence between the Clar  ...  The Clar covering polynomial (also called Zhang-Zhang polynomial in some chemical literature) of a hexagonal system is a counting polynomial for some types of resonant structures called Clar covers, which  ...  The resonance graph R(H) (also called Z-transformation graph) of a hexagonal system H was introduced independently by Gründler  , Zhang et al. [28, 29] and Randić [23, 24] .  ...

### Hexagonal systems with fixed bonds

Fuji Zhang, Xueliang Li, Heping Zhang
1993 Discrete Applied Mathematics
This decomposition can be used to simplify the procedure of finding Clar's formula, counting the number of Kekult structures and constructing the Z-transformation graph of a hexagonal system with fixed  ...  By this algorithm we can decompose a hexagonal system into a number of regions consisting of fixed bonds and a number of normal subhexagonal systems.  ...  Z-transformation graph for a hexagonal system with fixed bonds.  ...

### On plane bipartite graphs without fixed edges

Khaled Salem, Sandi Klavžar
2007 Applied Mathematics Letters
An edge of a graph H with a perfect matching is a fixed edge if it either belongs to none or to all of the perfect matchings of H .  ...  These results extend results on generalized hexagonal systems from [F. Zhang, M. Zheng, Generalized hexagonal systems with each hexagon being resonant, Discrete Appl. Math. 36 (1992) 67-73].  ...  Acknowledgements The first author thanks Professor Hernán Abeledo (Engineering Management and Systems Engineering, The George Washington University, USA) for drawing his attention to research on hexagonal  ...

### Asymptotic enumeration of perfect matchings in m-barrel fullerene graphs [article]

Afshin Behmaram, Cédric Boutillier
2017 arXiv   pre-print
In this paper we asymptotically count by two different methods the number of perfect matchings in m-barrel fullerene graphs, as the number of hexagonal layers is large, and show that the results are equal  ...  After that we have additional k layers of hexagons. At the last circle m-pentagons connected to the second m-gon.  ...  It is obvious that perfect matchings such a hexagonal graph on the cylinder with pentagonal faces on the boundary are in bijection with tilings of the cylinder with unit rhombi, where some of the rhombi  ...

### Page 59 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews
The Z-transformation graph Z(H) is the graph where the vertices are the perfect match- ings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of  ...  Michal Penn (IL-TECHMG; Haifa) 2000a:05164 05C70 05C90 Zhang, Lian Zhu (PRC-ZZTC; Zhangzhou) The Z-transformation graph of perfect matchings of a catacondensed hexagonal system. (Chinese.  ...
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