## On k-resonant fullerene graphs (2009)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Ye09onk-resonant,

author = {Dong Ye and Zhongbin Qi and Heping Zhang},

title = {On k-resonant fullerene graphs },

year = {2009}

}

### OpenURL

### Abstract

A fullerene graph F is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let M be a perfect matching of F. A cycle C of F is M-alternating if the edges of C appear alternately in and off M. A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that all hexagons in H are M-alternating. A fullerene graph F is k-resonant if any i (0 ≤ i ≤ k) disjoint hexagons of F form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2-resonant. Further, we show that a 3-resonant fullerene graph has at most 60 vertices and construct all nine 3-resonant fullerene graphs, which are also k-resonant for every integer k> 3. Finally, sextet polynomials of the 3-resonant fullerene graphs are computed.

### Citations

551 |
Matching Theory
- Lovasz, Plummer
- 1986
(Show Context)
Citation Context ...of G adjoins a subgraph G ′ of G if f is not a face of G ′ and f has an edge in common with G ′ . The faces adjoining G ′ are always called adjacent faces of G ′ . A subgraph H of G is called nice in =-=[20]-=- or central in [23] if G − V (H) has a perfect matching. So a resonant pattern of G can be viewed as a central subgraph of G. A graph G is cyclically k-edge connected if deleting fewer than k edges of... |

275 | Convex Polytopes
- Grünbaum
- 2003
(Show Context)
Citation Context ...graph is a 3-connected plane cubic graph with exactly 12 pentagonal faces and the other faces being hexagonal. Fullerene graphs have been studied in mathematics as trivalent polyhedra for a long time =-=[9, 12]-=-, for example, the dodecahedron is the fullerene graph with 20 vertices. Fullerene graphs have been studied in chemistry as fullerene molecules which have extensive applications in physics, chemistry ... |

57 | Permanents, Pfaffian orientations, and even directed circuits
- Robertson, Seymour, et al.
- 1999
(Show Context)
Citation Context ...raph G ′ of G if f is not a face of G ′ and f has an edge in common with G ′ . The faces adjoining G ′ are always called adjacent faces of G ′ . A subgraph H of G is called nice in [20] or central in =-=[23]-=- if G − V (H) has a perfect matching. So a resonant pattern of G can be viewed as a central subgraph of G. A graph G is cyclically k-edge connected if deleting fewer than k edges of G can not separate... |

45 |
The Aromatic Sextet
- Clar
- 1972
(Show Context)
Citation Context ...ng, where F − H denotes the subgraph obtained from F by deleting all vertices of H together with their incident edges. The maximum cardinality of resonant patterns of F is called the Clar number of F =-=[3]-=-, and the maximum number of M-alternating hexagons over all perfect matchings M of F is called the Fries number of F [8]. Graver [10] explored some connections among the Clar number, the face independ... |

28 |
A class of multi-symmetric polyhedra
- Goldberg
- 1937
(Show Context)
Citation Context ...graph is a 3-connected plane cubic graph with exactly 12 pentagonal faces and the other faces being hexagonal. Fullerene graphs have been studied in mathematics as trivalent polyhedra for a long time =-=[9, 12]-=-, for example, the dodecahedron is the fullerene graph with 20 vertices. Fullerene graphs have been studied in chemistry as fullerene molecules which have extensive applications in physics, chemistry ... |

10 |
Topological properties of benzenoid systems. Merrifield–Simmons indices and independence polynomials of unbranched catafusenes
- Gutman
- 1991
(Show Context)
Citation Context ...The sextet polynomial of C60 is computed [24] as BC60(x) = 5x 8 + 320x 7 + 1240x 6 + 1912x 5 + 1510x 4 + 660x 3 + 160x 2 + 20x + 1. (8) For a detailed discussion and review of sextet polynomials, see =-=[14, 22]-=-. Since any independent hexagons of a 3-resonant fullerene graph form a sextet pattern, we can compute easily the sextet polynomials of the other eight 3-resonant fullerene graphs as follows, by count... |

9 |
Aromaticity of Polycyclic Conjugated Hydrocarbons
- Randić
(Show Context)
Citation Context ...The sextet polynomial of C60 is computed [24] as BC60(x) = 5x 8 + 320x 7 + 1240x 6 + 1912x 5 + 1510x 4 + 660x 3 + 160x 2 + 20x + 1. (8) For a detailed discussion and review of sextet polynomials, see =-=[14, 22]-=-. Since any independent hexagons of a 3-resonant fullerene graph form a sextet pattern, we can compute easily the sextet polynomials of the other eight 3-resonant fullerene graphs as follows, by count... |

8 | Leapfrog Transformation and polyhedra of Clar Type
- PW, Pisanski
- 1994
(Show Context)
Citation Context ...ious that F −H is not bipartite. By Theorem 2.2 and Lemma 2.3, H is central. That means H is resonant. 3 2-resonant fullerene graphs Let F be a fullerene graph. The leapfrog operation on F is defined =-=[7]-=- as follows: for any face f of F, add a new vertex vf in f and join vf to all vertices in V (f) to obtain a new triangular graph F ′; then take the geometry dual of the graph F ′ and denote it by F ∗ ... |

7 |
On lower bounds of number of perfect matchings in fullerene graphs
- Doˇslić
- 1998
(Show Context)
Citation Context ...no. 10831001). † Corresponding author. 1A cycle C of G is M-alternating if the edges of C appear alternately in and off M. For a fullerene graph F, every edge of F belongs to a perfect matching of F =-=[16, 4]-=-. A hexagon h of a fullerene graph F is resonant if F has a perfect matching M such that h is M-alternating. It was proved that every hexagon of a normal benzenoid system is resonant [28]. This result... |

7 | Encoding fullerenes and geodesic domes
- Graver
(Show Context)
Citation Context ...e if it arises from a fullerene graph by the leapfrog operation. Several characterizations of leapfrog fullerenes have been given; see Liu, Klein and Schmalz [19], Fowler and Pisanski [7], and Graver =-=[10, 11]-=-. For example, a fullerene graph is a leapfrog fullerene if and only if it has a perfect Clar structure (i.e. a set of disjoint faces including all vertices); and if and only if it has a Fries structu... |

7 |
Sextet Polynomial: A New Enumeration and Proof Technique for Resonance Theory Applied to the Aromatic Hydrocarbons." Tetrahedron Letters
- Hosoya, Yamaguchi
- 1975
(Show Context)
Citation Context ...esonant for any integer k ≥ 3. 236 Sextet polynomials of 3-resonant fullerene graphs The sextet polynomial of a benzenoid system G for counting sextet patterns was introduced by Hosoya and Yamaguchi =-=[15]-=- as follows: BG(x) = C(G) ∑ i=0 σ(G, i)x i , (7) where σ(G, i) denotes the number of sextet patterns of G with i hexagons, and C(G) the Clar number of G. The sextet polynomial of C60 is computed [24] ... |

7 |
Cyclical edge-connectivity of fullerene graphs and (k
- Doˇslić
- 2003
(Show Context)
Citation Context ...dges out, contradicting the fact that every Gi sends precise three edges out. Therefore G − H is a bipartite graph with bipartition (S, G − H − S). This completes the proof of the theorem. Lemma 2.3. =-=[5, 16]-=- Every fullerene graph is cyclically 5-edge connected. By Lemma 2.3 and Theorem 2.2, we immediately have following result. Theorem 2.4. Every hexagon of a fullerene graph is resonant. Proof: Let F be ... |

6 |
Theorems of carbon cages
- Klein, Liu
- 1992
(Show Context)
Citation Context ...no. 10831001). † Corresponding author. 1A cycle C of G is M-alternating if the edges of C appear alternately in and off M. For a fullerene graph F, every edge of F belongs to a perfect matching of F =-=[16, 4]-=-. A hexagon h of a fullerene graph F is resonant if F has a perfect matching M such that h is M-alternating. It was proved that every hexagon of a normal benzenoid system is resonant [28]. This result... |

5 |
On cyclic edge-connectivity of fullerenes
- Kutnar, Maruˇsič
(Show Context)
Citation Context ...cle and lying on the outer cycle respectively, which would result in one face of size at most four in F, a contradiction. Hence τ(F) ≥ 5. The following lemma is due to Kutnar and Maruˇsič. Lemma 4.6. =-=[17]-=- Let F be a fullerene graph containing a polygonal ring R of length five, and let C and C ′ be the inner cycle and the outer cycle of R, respectively. Then either (1) C or C ′ is the boundary of a fac... |

5 |
k-resonance in toroidal polyhexes
- Shiu, Lam, et al.
- 2005
(Show Context)
Citation Context ...systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], open-ended nanotubes [29], toroidal polyhexes =-=[25, 32]-=- and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene graphs. We show that all leapfrog fullerene graphs are 2-res... |

5 |
When each hexagon of a hexagonal system covers it
- Zhang, Chen
- 1991
(Show Context)
Citation Context ...hing of F [16, 4]. A hexagon h of a fullerene graph F is resonant if F has a perfect matching M such that h is M-alternating. It was proved that every hexagon of a normal benzenoid system is resonant =-=[28]-=-. This result was generalized to normal coronoid systems [30] and plane elementary bipartite graphs [33]. However a fullerene graph is a non-bipartite graph. It is natural to ask if every hexagon of a... |

5 |
Generalized hexagonal systems with each hexagon being resonant
- Zhang, Zheng
- 1992
(Show Context)
Citation Context ...nant if F has a perfect matching M such that h is M-alternating. It was proved that every hexagon of a normal benzenoid system is resonant [28]. This result was generalized to normal coronoid systems =-=[30]-=- and plane elementary bipartite graphs [33]. However a fullerene graph is a non-bipartite graph. It is natural to ask if every hexagon of a fullerene graph is resonant. The present paper first uses Tu... |

5 |
Plane elementary bipartite graphs
- Zhang, Zhang
- 2000
(Show Context)
Citation Context ...t h is M-alternating. It was proved that every hexagon of a normal benzenoid system is resonant [28]. This result was generalized to normal coronoid systems [30] and plane elementary bipartite graphs =-=[33]-=-. However a fullerene graph is a non-bipartite graph. It is natural to ask if every hexagon of a fullerene graph is resonant. The present paper first uses Tutte’s 1-factor theorem to give a positive a... |

4 |
Kekulé structures and the face independence number of a fullerene
- Graver
(Show Context)
Citation Context ...ardinality of resonant patterns of F is called the Clar number of F [3], and the maximum number of M-alternating hexagons over all perfect matchings M of F is called the Fries number of F [8]. Graver =-=[10]-=- explored some connections among the Clar number, the face independence number and the Fries number of a fullerene graph, and obtained a lower bound for the Clar number of leapfrog fullerene graphs wi... |

4 |
k-resonance of open-ended carbon nanotubes
- Zhang, Wang
- 2004
(Show Context)
Citation Context ...ral k-resonant benzenoid systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], open-ended nanotubes =-=[29]-=-, toroidal polyhexes [25, 32] and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene graphs. We show that all leapfr... |

3 |
Preferable fullerenes and Clar-sextet cages, Full
- Liu, Klein, et al.
- 1994
(Show Context)
Citation Context ...ullerene graph is called leapfrog fullerene if it arises from a fullerene graph by the leapfrog operation. Several characterizations of leapfrog fullerenes have been given; see Liu, Klein and Schmalz =-=[19]-=-, Fowler and Pisanski [7], and Graver [10, 11]. For example, a fullerene graph is a leapfrog fullerene if and only if it has a perfect Clar structure (i.e. a set of disjoint faces including all vertic... |

3 |
A note on the cyclical edge-connectivity of fullerene graphs
- Qi, Zhang
- 2003
(Show Context)
Citation Context ..., contradicting the fact that every Gi sends precisely three edges out. 3Therefore G − H is a bipartite graph with bipartition (S, V (G − H − S)). This completes the proof of the theorem. Lemma 2.3. =-=[5, 21]-=- Every fullerene graph is cyclically 5-edge connected. By Lemma 2.3 and Theorem 2.2, we immediately have the following result. Theorem 2.4. Every hexagon of a fullerene graph is resonant. Proof: Let F... |

3 |
An upper bound for the Clar number of fullerene graphs
- Zhang, Ye
- 2007
(Show Context)
Citation Context ...lar number, the face independence number and the Fries number of a fullerene graph, and obtained a lower bound for the Clar number of leapfrog fullerene graphs with icosahedral symmetry. Zhang and Ye =-=[31]-=- showed that the Clar number of a fullerene graph Fn with n vertices satisfies c(Fn) ≤ ⌊n−12 ⌋, which is sharp for infinitely 6 many fullerene graphs, including C60 whose Clar number is 8 [1]. Shiu, L... |

3 |
Clar and sextet polynomials of buckminsterfullerene
- Shiu, Lam, et al.
(Show Context)
Citation Context ... that Clar number of a fullerene graph Fn with n vertices satisfies c(Fn) ≤ ⌊ n−12 6 ⌋ which is sharp for infinite many fullerene graphs, including C60 whose Clar number is 8 [1]. Shiu, Lam and Zhang =-=[18]-=- computed the Clar polynomial and sextet polynomial of C60 by showing that every face independent set of C60 is also a resonant pattern. A fullerene graph is k-resonant if any i (0 ≤ i ≤ k) disjoint h... |

2 |
Clar sextet theory of buckminsterfullerene (C60
- El-Basil
- 2000
(Show Context)
Citation Context ... and Ye [31] showed that the Clar number of a fullerene graph Fn with n vertices satisfies c(Fn) ≤ ⌊n−12 ⌋, which is sharp for infinitely 6 many fullerene graphs, including C60 whose Clar number is 8 =-=[1]-=-. Shiu, Lam and Zhang [24] computed the Clar polynomial and the sextet polynomial of C60 by showing that every hexagonal face independent set of C60 is also a resonant pattern. A fullerene graph is k-... |

2 |
k-coverable coronoid systems
- Chen, Guo
- 1993
(Show Context)
Citation Context ...ng [34, 35] characterized general k-resonant benzenoid systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems =-=[2, 18]-=-, open-ended nanotubes [29], toroidal polyhexes [25, 32] and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene grap... |

2 |
An Atlas of Fullerenes (Oxford Univ
- Fowler, Manolopoulos
- 1995
(Show Context)
Citation Context ...decahedron is the fullerene graph with 20 vertices. Fullerene graphs have been studied in chemistry as fullerene molecules which have extensive applications in physics, chemistry and material science =-=[6]-=-. Let G be a plane 2-connected graph. A perfect matching or 1-factor M of G is a set of independent edges such that every vertex of G is incident with exactly one edge in M. ∗This work is supported by... |

2 |
Bicyclic compounds and their comparison with naphthalene
- Fries
- 1972
(Show Context)
Citation Context ...he maximum cardinality of resonant patterns of F is called the Clar number of F [3], and the maximum number of M-alternating hexagons over all perfect matchings M of F is called the Fries number of F =-=[8]-=-. Graver [10] explored some connections among the Clar number, the face independence number and the Fries number of a fullerene graph, and obtained a lower bound for the Clar number of leapfrog fuller... |

2 |
k-resonant benzenoid systems and k-cycle resonant graphs
- Guo
(Show Context)
Citation Context ...s result also holds for coronoid systems [2, 18], open-ended nanotubes [29], toroidal polyhexes [25, 32] and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to =-=[13]-=-. Here we consider k-resonant fullerene graphs. We show that all leapfrog fullerene graphs are 2-resonant and a 3-resonant fullerene graph has at most 60 vertices. We construct all 3-resonant fulleren... |

2 |
k-resonant toroidal polyhexes
- Zhang, Ye
(Show Context)
Citation Context ...systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], open-ended nanotubes [29], toroidal polyhexes =-=[25, 32]-=- and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene graphs. We show that all leapfrog fullerene graphs are 2-res... |

2 |
k-resonant benzenoid systems
- Zheng
- 1991
(Show Context)
Citation Context ...≤ k) disjoint hexagons are mutually resonant. So k-resonant fullerene graphs are also (k − 1)-resonant for integer k ≥ 1. Hence a fullerene graph with each hexagon being resonant is 1-resonant. Zheng =-=[34, 35]-=- characterized general k-resonant benzenoid systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], op... |

2 |
Construction of 3-resonant benzenoid systems
- Zheng
- 1992
(Show Context)
Citation Context ...≤ k) disjoint hexagons are mutually resonant. So k-resonant fullerene graphs are also (k − 1)-resonant for integer k ≥ 1. Hence a fullerene graph with each hexagon being resonant is 1-resonant. Zheng =-=[34, 35]-=- characterized general k-resonant benzenoid systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], op... |

1 |
k-coverable polyhex graphs, Ars Combin
- Lin, Chen
- 1996
(Show Context)
Citation Context ...ng [34, 35] characterized general k-resonant benzenoid systems. In particular, he showed that every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems =-=[2, 18]-=-, open-ended nanotubes [29], toroidal polyhexes [25, 32] and Klein-bottle polyhexes [26]. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene grap... |

1 |
A complete charaterization for k-resonant Klein-bottle polyhexes
- Shiu, Zhang
- 2008
(Show Context)
Citation Context ...at every 3-resonant benzenoid system is also k-resonant (k ≥ 3). This result also holds for coronoid systems [2, 18], open-ended nanotubes [29], toroidal polyhexes [25, 32] and Klein-bottle polyhexes =-=[26]-=-. For a recent survey on k-resonant benzenoid systems, refer to [13]. Here we consider k-resonant fullerene graphs. We show that all leapfrog fullerene graphs are 2-resonant and a 3-resonant fullerene... |

1 | Extremal fullerene graphs with the maximum Clar number, submitted
- Ye, Zhang
(Show Context)
Citation Context ...ns. For a pentagonal fragment B, use γ(B) denote the minimum number of pentagons adjoining a pentagon in B. For example, γ(R5) = 3. The following two lemmas due to Ye and Zhang are useful. Lemma 4.2. =-=[27]-=- Let B be a fragment of a fullerene graph F and W the set of 2-degree vertices on the boundary ∂B. If 0 < |W | ≤ 4, then T = F − (V (B) \ W) is a forest and (1) T is K2 if |W | = 2; (2) T is K1,3 if |... |