The shape of a single-walled carbon nanotube’s cylinder is described by its chiral indices, (n,m), and important properties of the nanotube are determined by this pair of values. In particular, a ...nanotube is metallic or quasi-metallic when n-m is a multiple of 3, and is otherwise a semiconductor. This paper characterizes the conjugated π-systems that can form on capped nanotubes in each case. When n-m is a multiple of 3, there is a fully conjugated π-system running along the nanotube’s cylinder such that two-thirds of the hexagons are benzene rings and one-third are in a resonant set. In contrast, when n-m is not a multiple of 3, the pattern is broken along the length of the cylinder by two fracture lines. Surprisingly, and contrary to conventional thinking, these results are completely independent of the nanotube caps. The results are related to a similar characterization for open nanotubes by Ormsby and King, although in that case only a single fracture line is necessary. This new work is backed by the authors’ previous results on the Clar numbers of fullerenes in general. It also provides new predictions for nanotubes and introduces the concept of the aromaticity ratio.
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•Dominant resonance structure of an (n,m) nanotube is given by the value n-m modulo 3.•When the value is 0, the nanotube is metallic and the cylinder is fully conjugated.•Otherwise, the nanotube is semiconducting with two fracture lines of double bonds.•Surprisingly, nanotube caps do not affect the valence bond structure of its cylinder.•We use results on the Clar number of fullerenes and introduce the aromaticity ratio.
Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are ...partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality.
Given the Kekulé structure of a fullerene that gives its Clar structure, the Kekulé edges that do not lie on any benzene ring of the Clar structure lie on open chains that pair up the pentagonal ...faces and perhaps some closed chains. In this paper, we introduce Fries chains and show that given the Kekulé structure of a fullerene that gives its Fries structure, the Kekulé edges that do not lie on two benzene rings lie on the union of a set of these Fries chains. The edges that lie on exactly one benzene ring belong to exactly one of these chains while edges that lie on no benzene ring belong to exactly two of these chains. We will see that the Fries chains will include the Clar chains in some cases and will be quite distinct from the Clar chains in other cases.
Examples of fullerenes with the property that the set of benzene rings that give its Clar number are not a subset of the benzene rings that give its Fries number have been known for a few years. However, in all of these known examples, almost all of the benzene rings of its Clar structure are among benzene rings of its Fries structure. In this paper, we describe a class of fullerenes with the property that the set of benzene rings of its Clar structure and the set of benzene rings of its Fries structure practically disjoint.
Numbers of faces in disordered patches Brinkmann, Gunnar; Graver, Jack E.; Justus, Claudia
Journal of mathematical chemistry,
02/2009, Letnik:
45, Številka:
2
Journal Article, Conference Proceeding
Recenzirano
Odprti dostop
It has been shown that the boundary structure of patches with all faces of the same size
k
, all interior vertices of the same degree
m
and all boundary vertices of degree at most
m
determines the ...number of faces of the patch (Brinkmann et al., Graphs and discovery, 2005; Guo et al., Discrete Appl Math 118(3):209–222, 2002). In case of at least two defective faces, that is faces with degree
k
′ ≠
k
, it is well known that this is not the case. The most famous example for this is the Endo–Kroto
C
2
-insertion (Endo and Kroto, J Phys Chem 96:6941–6944, 1992). Patches with alimited amount of
disorder
are especially interesting for the case
k
= 6,
m
= 3 and
k
′ = 5. This case corresponds to polycyclic hydrocarbons with a limited number of pentagons and to subgraphs of fullerenes. The last open question was the case of exactly one defective face or vertex. In this paper we generalize the results of Brinkmann et al. (2005) and Guo et al. (2002) and in some cases corresponding to Euclidean lattices also deal with patches that have vertices of degree larger than
m
on the boundary, have sequences of degrees on the boundary that are identical only modulo
m
and have vertex and face degrees in the interior that are multiples of
m
, resp.
k
. Furthermore we prove that in case of at most one defective face with a degree that is not a multiple of
k
the number of faces of a patch is determined by the boundary. This result implies that fullerenes cannot grow by replacing patches of a restricted size.
A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product. Hence, the proof that dim(S) + dim(S
⊥
) = dim(ν) ...is not presented in a way that generalizes to non-positive-definite inner products or to vector spaces over other fields. In this note we give such a proof.
WHEN DOES A CURVE BOUND A DISTORTED DISK? GRAVER, Jack E; CARGO, Gerald T
SIAM journal on discrete mathematics,
2011, 2011-01-00, 20110101, Letnik:
25, Številka:
1-2
Journal Article
Recenzirano
Consider a closed curve in the plane that does not intersect itself; by the Jordan-Schoenflies theorem, it bounds a distorted disk. Now consider a closed curve that intersects itself, perhaps several ...times. Is it the boundary of a distorted disk that overlaps itself? If it is, is that distorted disk essentially unique? In this paper, we develop techniques for answering both of these questions for any given closed curve in the plane. PUBLICATION ABSTRACT
Celotno besedilo
Dostopno za:
CEKLJ, DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
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