k-resonance in Toroidal Polyhexes*

Shiu, Wai Chee; Lam, Peter Che Bor; Zhang, Heping

doi:10.1007/s10910-004-6897-4

Cited by 24 publications

(8 citation statements)

References 25 publications

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“…A toroidal polyhex H ( p , q , t ) arises from P by identifying the lateral sides, then bottom and top cycles after t torsion (for details, refer to Ref. 31). Hence M 7 can be drawn as H (7,1,3) in Figure 3.…”

Section: Examplesmentioning

confidence: 99%

Resistance distance and Kirchhoff index in circulant graphs

Zhang

Yang

2006

Int J of Quantum Chemistry

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113

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ABSTRACT:The resistance distance r ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf (G) is the sum of resistance distances between all pairs of vertices. In this work, closed-form formulae for Kirchhoff index and resistance distances of circulant graphs are derived in terms of Laplacian spectrum and eigenvectors. Special formulae are also given for four classes of circulant graphs complete graphs, complete graphs minus a perfect matching, cycles, Möbius ladders M p . In particular, the asymptotic behavior of Kf

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Section: Examplesmentioning

confidence: 99%

Resistance distance and Kirchhoff index in circulant graphs

Zhang

Yang

2006

Int J of Quantum Chemistry

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113

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“…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].…”

Section: Introductionmentioning

confidence: 75%

Onk-Resonant Fullerene Graphs

Ye¹,

Zhong-bin

Zhang³

2009

SIAM J. Discrete Math.

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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.

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“…In [22], [30], remarkable work have been done in the context of full identification of toroidal fullerenes by employing diversified procedure. Some results about k-resonance of toroidal fullerenes and Kleinbottle polyhex can be found in [31]- [33]. 2-extendability of toroidal and Klein-bottle polyhexes is explained in [34].…”

Section: Graphical Identifications Of Toroidal and Klein-bottle Fmentioning

confidence: 88%

Cyclic Super Magic Labelings for Toroidal and Klein-Bottle Fullerenes

Numan

Butt

et al. 2019

IEEE Access

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In the present paper, we investigate the cyclic-supermagic behavior of toroidal and Klein-bottle graph.INDEX TERMS Toroidal fullerene, Klein-bottle fullerene, toroidal polyhex, Klein-bottle polyhex, cyclic supermagic labeling. I. INTRODUCTION AND DEFINITIONSFullerenes, the third type of carbon, have ended up essential atoms in science and innovation. In [1], [6], a massive literature is discussed about fullerenes along with their applications in science and innovation. Fullerenes are atoms made altogether out of carbon that were found in 1985 at Rice University. A large number of the viable uses of fullerenes take after straightforwardly from their uncommon properties. Significant chemical, physical and optical properties have been discussed in [7], which makes fullerenes key segments for the eventual fate of nano-electromechanical frameworks. The most significant properties are displayed as its high electron affinity and oxidation of the atom. Fullerenes are amazingly solid atoms, ready to oppose extraordinary weights. In optical properties fullerenes have indicated specific guarantees in optical restricting and power subordinate refractive list. The uses of fullerene are utilizations in solar cell, hydrogen gas storage devices, harden metals and alloys, interdigitated capacitors (IDCs), treatment of AIDS and also in magnetic resonance imaging (MRI).Let G = (V (G), E(G)) be a finite simple graph, where V (G) denote vertex set and E(G) denote edge set of G. The cardinality of vertices is denoted by |V (G)| and cardinality of edges denoted by |E(G)| respectively.The associate editor coordinating the review of this manuscript and approving it for publication was Luca Barletta.Let K 1 , K 2 , . . . , K t be a collection of subgraphs of G then G has (K 1 , K 2 , . . . , K t )-edge-covering, if every edge of E(G) belongs to a subgraph K i , 1 ≤ i ≤ t. If each subgraph K i is isomorphic to some graph K , then G admits K -covering.Consider G has K -covering. A bijective function f from the setThe weight of a subgraph K , of G under f is the sum of vertices labels and edges labels correspond to K , . The total labeling f is a K -magic labeling of G if the weight of each subgraph isomorphic to K is equal to a given constant C. The graph G is K -magic if G has K -magic labeling. Moreover, if we used smallest possible labels for vertices of G, that is if f (v) ∈ {1, 2, 3, . . . , |V (G)|}, then G is said to be H -supermagic.In 2005 Gutierrez and Llado [18] first introduced the idea of H -supermagic labeling. In [18] they showed that for some values of n the complete bipartite graph K n,m are K 1,n -supermagic. They also showed that for some n the path P n and the cycle C n are P h -supermagic. Later in 2007 Llado and Moragas [23] described some C n -super-magic graphs. Jeyanthi and Selvagopal [27], [29] discussed the H -super magic strength of graphs and some C 4 -super magic graphs moreover some results of C 3 -super magic graphs can be found in [28]. In [24] the author showed that if a graph is C-supermagic then disjoint ...

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k-resonance in Toroidal Polyhexes*

Cited by 24 publications

References 25 publications

Resistance distance and Kirchhoff index in circulant graphs

Resistance distance and Kirchhoff index in circulant graphs

Onk-Resonant Fullerene Graphs

Cyclic Super Magic Labelings for Toroidal and Klein-Bottle Fullerenes

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