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Fullerenes as Tilings of Surfaces

2000
*
Journal of chemical information and computer sciences
*

If a fullerene is defined as a finite trivalent

doi:10.1021/ci990066h
pmid:10850758
fatcat:v5y7h5jb4zaqvinyud5ennwfim
*graph*made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane ... Hence, each*K*2 -*polyhex*on n vertices is doubly*covered*by a centrosymmetric T 2 -*polyhex*on 2n vertices and is therefore a diVisor 42 of the larger*graph*. ...*K*2 -*polyhex*on 24 vertices,*covered*by a 48-vertex, centrosymmetric D 6h torus. ...##
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Page 4717 of Mathematical Reviews Vol. , Issue 97H
[page]

1997
*
Mathematical Reviews
*

In this paper, the authors give a constructive criterion to determine whether or not a given

*polyhex**graph*is*k*-*coverable*. ... Roditty (IL-TLAV; Tel Aviv) 05C*Graph*theory 97h:05138 05C70 Lin, Ke-rong (PRC-FZHU; Fuzhou); Chen, Rong-si (PRC-FZHU-FE; Fuzhou)*k*-*coverable**polyhex**graphs*. ...##
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On the SD-Polynomial and SD-Index of an Infinite Class of "Armchair Polyhex Nanotubes"

2014
*
International Letters of Chemistry, Physics and Astronomy
*

Let G be a simple connected

doi:10.18052/www.scipress.com/ilcpa.31.63
fatcat:3tinfbotwrdaxfjktjp5skpoui
*graph*with the vertex set V = V(G) and the edge set E = E(G),without loops and multiple edges. ... .,*k*). One can obtain the Sd-polynomial by replacing xc with x|E(G)|-c in Ω-polynomial. Then the Sd-index will be the first derivative of Sd(x) evaluated at x = 1. ... The*graph*G is called co-*graph*if and only if the edge set E(G) a union of disjoint orthogonal cuts: E(G) = 1*k*i C i and C i ∩C j = , for i≠j and i,j = 1, 2, …,*k*. ...##
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Resonance in elemental benzenoids

1996
*
Discrete Applied Mathematics
*

Relevance is also indicated for a class of benzenoid structures corresponding to cylinders with

doi:10.1016/0166-218x(95)00017-l
fatcat:irbtsez7vnevxhpqq7k6apbikq
*polyhex**coverings*on the sides (but not the "ends"). * Corresponding author. 0166-218X/96/$15.00 0 1996 Elsevier ... This enumeration is analytically performed here for the finite-sized elemental benzenoid*graphs*corresponding to hexagonal*coverings*on suitable closed surfaces, namely the torus and the Klein bottle. ... Conclusion Kekult structures (or perfect matchings) have been analytically enumerated for a variety of*polyhex**covering**graphs*on closed surfaces. ...##
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Π(G,x) Polynomial and Π(G) Index of Armchair Polyhex Nanotubes TUAC6[m,n]

2014
*
International Letters of Chemistry, Physics and Astronomy
*

Let G be a simple connected

doi:10.18052/www.scipress.com/ilcpa.36.201
fatcat:4uwoeako7bfync5sf6z3mafbom
*graph*with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. ... defined by Ashrafi et al as Sd(G,x) = ∑cm(G,c)x[E(G)]-c in this paper we compute the Pi polynomial Π(G,x) = ∑cm(G,c)x[E(G)]-c and Pi Index Π(G ) = ∑cc·m(G,c)([E(G)]-c) of an infinite class of "Armchair*polyhex*... Two edges e = uv and f = xy of G e co f if and only if d(u,x) = d(v,y) =*k*and d(u,y) = d(v,x) =*k*+1 or vice versa, for a nonnegative integer*k*. ...##
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Non saturated polyhexes and polyiamonds

2018
*
International Conference on Random and Exhaustive Generation of Combinatorial Structures
*

After an overview of this recent subject, we proceed to the enumeration of fixed non saturated

dblp:conf/gascom/MasseCG18
fatcat:wlf7nubzoneajkbmgnaiedlvqe
*polyhexes*and polyiamonds. ... Induced subtrees of a*graph*G are induced subgraphs of G that are trees. Fully leafed induced subtrees of G have the maximal number of leaves among all induced subtrees of G. ... These*cover*all cases in the set C t (0)C t (*k*) and we have c t (0)c t (*k*) = 60 + 6k for*k*≥ 5. ...##
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Page 3443 of Mathematical Reviews Vol. , Issue 97F
[page]

1997
*
Mathematical Reviews
*

Xiaoya Zha (1-VDB; Nashville, TN)
97f:05186 05C90 05C70
Chen, Rong-si (PRC-FZHU-FE; Fuzhou)

*k*-*coverable**polyhexes*. ... A*polyhex*is said to be*k*-*coverable*if for any ¢ (<*k*) pairwise disjoint hexagons the subgraph obtained from the*polyhex*by deleting all these ¢ hexagons together with their incident edges has at least ...##
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2-extendability of toroidal polyhexes and Klein-bottle polyhexes

2009
*
Discrete Applied Mathematics
*

A connected

doi:10.1016/j.dam.2008.03.009
fatcat:hoessqshynhrji3ra7s2uzx4iy
*graph*G admitting a perfect matching is kextendable if |V(G)| ≥ 2k + 2 and any*k*independent edges can be extended to a perfect matching of G. ... In this paper, we characterize 2-extendable toroidal*polyhexes*and 2extendable Klein-bottle*polyhexes*. ... Toroidal*polyhexes*are bipartite and*cover*many interesting*graphs*, such as*K*3,3 , Cube (Q 3 ), Heawood*graph*, generalized Petersen*graph*G (8, 3) [15, 24] and some circulant*graphs*[29] . ...##
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k-resonance in Toroidal Polyhexes*

2005
*
Journal of Mathematical Chemistry
*

A toroidal

doi:10.1007/s10910-004-6897-4
fatcat:ous4cp4i35hbflig3tkebqryxa
*polyhex*G is said to be*k*-resonant if, for 1 ≤ i ≤*k*, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M -alternating ... This paper considers the*k*-resonance of a toroidal*polyhex*(or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). ... A benzenoid system is said to be*k*-resonant if, for 1 ≤ i ≤*k*, any i disjoint hexagons are mutually resonant. For a recent survey on*k*-resonant benzenoids and*k*-cycle resonant*graphs*, see [5] . ...##
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"Pure-Polyhex" π-Networks: Topo-Combinatorics

2021
*
Croatica Chemica Acta
*

These are viewed as hexagonal-network

doi:10.5562/cca3790
fatcat:e2svrumekva7vk5puqnj3yryn4
*coverings*(i.e., a tiling by hexagons) of a connected locally Euclidean surface S possibly with holes which can be simple cycles of sizes other than 6. ... Structural possibilities are considered for what arguably is the most general class of connected "pure-*polyhex*" π-networks (of carbon atoms). ... δ*K*G π π π = = = . ⯀ Theorem 13: Let G be a pure*polyhex*species, with a finite untwisted circuit boundary δG , with a hole*graph*hole G . ...##
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Omega polynomial in twisted/chiral polyhex tori

2008
*
Journal of Mathematical Chemistry
*

The newly proposed Omega counting polynomial is investigated in case of twisted/chiral

doi:10.1007/s10910-008-9407-2
fatcat:ddskyu534ndhtlapwz3vd2lygu
*polyhex*tori. Basic definition and properties are given. ... A factorization procedure is proposed for finding analytical relationships in families of*polyhex*toroidal nanostructures. Numerical data are also given. ... The coefficients m(G,*k*) in the polynomial expression: (2) can be calculated from the*graph*G by a method making use of the Sachs*graphs*, which are subgraphs of G. ...##
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2-extendability and k-resonance of non-bipartite Klein-bottle polyhexes

2011
*
Discrete Applied Mathematics
*

Further, we completely characterize 2-extendable and

doi:10.1016/j.dam.2010.12.015
fatcat:pptg36upljgdfiafdi2rrwmagu
*k*-resonant non-bipartite Klein-bottle*polyhexes*, respectively. ... Thomassen classified Klein-bottle*polyhexes*into five classes [C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive*graphs*on a fixed surface, Trans. Amer. Math. ... N(p, q) (q ≥ 2) is not 3-extendable by the fact that a*k*-extendable*graph*is (*k*+ 1)-connected. We now consider the 2-extendability of the non-bipartite Klein-bottle*polyhexes*. ...##
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The electronic structure of polyhex carbon tori

2000
*
Journal of Chemical Physics
*

condition: a metallic

doi:10.1063/1.480972
fatcat:5q2g6ts5uben3h4ezn2e2wwsy4
*polyhex*torus is constructible as a leapfrog transformation of a smaller*polyhex*torus. ... Vanishing of the gap between highest occupied and lowest unoccupied molecular orbitals for a torus imposes divisibility by 3 on the indices of chiral and twisting vectors, which translates into one*graph*-theoretical ... The authors thank*K*. M. Rogers for help in preparing some of the figures. ...##
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Omega Polynomial in Nanostructures
[chapter]

2016
*
Distance, Symmetry, and Topology in Carbon Nanomaterials
*

Closed analytical formulas for W(G, x) and CI in

doi:10.1007/978-3-319-31584-3_2
fatcat:u4jhpkytqbeavgyy2fkfuqgahq
*polyhex*tori and tubes are given. ... A new counting polynomial, called the »Omega« W(G, x) polynomial, was recently proposed by Diudea on the ground of quasi-orthogonal cut »qoc« edge strips in a polycyclic*graph*. ...*Graph*G is called a co-*graph*if and only if the edge set E(G) is the union of disjoint orthogonal cuts: C 1 ∪ C 2 ∪ ... ∪ C*k*= E and C i ∩ C j = Ø for i ≠ j, i, j = 1,2,...,*k*. ...##
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Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems

2020
*
Symmetry
*

Moreover, this newly conjectured topological similarity between the two kinds of

doi:10.3390/sym12081233
fatcat:allxmdnmb5eg3dpycf6xobczna
*graphs*transfers the translation invariance typical of the graphenic tori to the Klein bottle*polyhexes*with size LC ≥ LC ... , making these*graphs*vertex transitive. ... will change in such a way that all the nodes of both*graphs*show the same eccentricity ε = ε and the same set of coordination numbers {b*k*} with*k*= 1,2,.. ...
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