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Fullerenes as Tilings of Surfaces

M. Deza, P. W. Fowler, A. Rassat, K. M. Rogers
2000 Journal of chemical information and computer sciences  
If a fullerene is defined as a finite trivalent 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.  ... 
doi:10.1021/ci990066h pmid:10850758 fatcat:v5y7h5jb4zaqvinyud5ennwfim

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.  ... 

On the SD-Polynomial and SD-Index of an Infinite Class of "Armchair Polyhex Nanotubes"

Mohammad Reza Farahani
2014 International Letters of Chemistry, Physics and Astronomy  
Let G be a simple connected 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.  ... 
doi:10.18052/www.scipress.com/ilcpa.31.63 fatcat:3tinfbotwrdaxfjktjp5skpoui

Resonance in elemental benzenoids

D.J. Klein, H. Zhu
1996 Discrete Applied Mathematics  
Relevance is also indicated for a class of benzenoid structures corresponding to cylinders with 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.  ... 
doi:10.1016/0166-218x(95)00017-l fatcat:irbtsez7vnevxhpqq7k6apbikq

Π(G,x) Polynomial and Π(G) Index of Armchair Polyhex Nanotubes TUAC6[m,n]

Mohammad Reza Farahani
2014 International Letters of Chemistry, Physics and Astronomy  
Let G be a simple connected 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.  ... 
doi:10.18052/www.scipress.com/ilcpa.36.201 fatcat:4uwoeako7bfync5sf6z3mafbom

Non saturated polyhexes and polyiamonds

Alexandre Blondin Massé, Julien de Carufel, Alain Goupil
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 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.  ... 
dblp:conf/gascom/MasseCG18 fatcat:wlf7nubzoneajkbmgnaiedlvqe

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  ... 

2-extendability of toroidal polyhexes and Klein-bottle polyhexes

Dong Ye, Heping Zhang
2009 Discrete Applied Mathematics  
A connected 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] .  ... 
doi:10.1016/j.dam.2008.03.009 fatcat:hoessqshynhrji3ra7s2uzx4iy

k-resonance in Toroidal Polyhexes*

Wai Chee Shiu, Peter Che Bor Lam, Heping Zhang
2005 Journal of Mathematical Chemistry  
A toroidal 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] .  ... 
doi:10.1007/s10910-004-6897-4 fatcat:ous4cp4i35hbflig3tkebqryxa

"Pure-Polyhex" π-Networks: Topo-Combinatorics

Douglas J. Klein, Bholanath Mandal
2021 Croatica Chemica Acta  
These are viewed as hexagonal-network 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 .  ... 
doi:10.5562/cca3790 fatcat:e2svrumekva7vk5puqnj3yryn4

Omega polynomial in twisted/chiral polyhex tori

Mircea V. Diudea
2008 Journal of Mathematical Chemistry  
The newly proposed Omega counting polynomial is investigated in case of twisted/chiral 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.  ... 
doi:10.1007/s10910-008-9407-2 fatcat:ddskyu534ndhtlapwz3vd2lygu

2-extendability and k-resonance of non-bipartite Klein-bottle polyhexes

Qiuli Li, Saihua Liu, Heping Zhang
2011 Discrete Applied Mathematics  
Further, we completely characterize 2-extendable and 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.  ... 
doi:10.1016/j.dam.2010.12.015 fatcat:pptg36upljgdfiafdi2rrwmagu

The electronic structure of polyhex carbon tori

A. Ceulemans, L. F. Chibotaru, S. A. Bovin, P. W. Fowler
2000 Journal of Chemical Physics  
condition: a metallic 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.  ... 
doi:10.1063/1.480972 fatcat:5q2g6ts5uben3h4ezn2e2wwsy4

Omega Polynomial in Nanostructures [chapter]

Mircea V. Diudea, Beata Szefler
2016 Distance, Symmetry, and Topology in Carbon Nanomaterials  
Closed analytical formulas for W(G, x) and CI in 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.  ... 
doi:10.1007/978-3-319-31584-3_2 fatcat:u4jhpkytqbeavgyy2fkfuqgahq

Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems

Mihai V. Putz, Ottorino Ori
2020 Symmetry  
Moreover, this newly conjectured topological similarity between the two kinds of 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,..  ... 
doi:10.3390/sym12081233 fatcat:allxmdnmb5eg3dpycf6xobczna
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