Boundary uniqueness of fusenes

It is shown that a geometrically planar fusene is uniquely determined by its boundary edge code. Surprisingly, the same conclusion is not true in general but holds for geometrically planar and non-planar fusenes with at most 25 hexagons, except for two particular cases. In addition, it is proved that two fusenes with the same boundary edge code have the same number of hexagons. (C) 2002 Elsevier Science B.V. All rights reserved

Application of the Dualist Model. Generation of Kekule Structures and Resonant Sextets of Benzenoid Hydrocarbons

The dualist model of Balaban is used for the enumeiration and dis.play of Kelmle struotmres K aind .resooant sextet numbers rCG; k) O·f la.rge cata-condenrsed bemmnoi:d hydrncarbons. The key steps are: (1) Transfo.rm the berw;enoid graph into the corresponding dualist and associate with it a linear-angular, L-A, sequence, (2) Fragment the dualist into subgraphs after each L-A pair. The resulting subgraphs are called fragment graiphs. (3) Colour each fragment graph, containing v veritices, v + 1 times such ·that each co1ourin.g contaiins a.t mo&t on black vertex (the rest being white). ( 4) Re-assemble the coloured fragments into theiir initial geometry, preserved in the dual\u27ist, to produce a set o.f c10l0>ured dua!ists such that no coloured dualist has more than one black vertex in each linear segment. The number of such coloured duali:sts is K, the Kek•ule count. By convention, each blac.k dualist vertex corresponds to a propex resona,nt sextet. This, plus the fac.t that a lineair segment can have at most one resonant sexteit, completely defines all o.f the individual VB Kekule structures and their resonant sextets. The method is an illustration o.f data reduction schemes and is quite suited for large benzenoid hydrocarbons. A numbe•r O<f fo:rlIIlllllae for com,putiing the number of Kekule striuctures of vairious families of cata-condensed benzenoid hydrocarbons are derived. In addition, the abnve apprnach is applicable to large benzenoid systems consisting 0if c.ata- condensed f.ra,gments and thin peri-condensed fragments

Novamene: A new class of carbon allotropes

We announce a new class of carbon allotropes. The basis of this new classification resides on the concept of combining hexagonal diamond (sp3 bonded carbon - lonsdaleite) and ring carbon (sp2 bonded carbon - graphene). Since hexagonal diamond acts as an insulator and sp2 bonded rings act as conductors, these predicted materials have potential applications for transistors and other electronic components. We describe the structure of a proposed series of carbon allotropes, novamene, and carry out a detailed computational analysis of the structural and electronic properties of the simplest compound in this class: the single-ring novamene. In addition, we suggest how hundreds of different allotropes of carbon could be constructed within this class

Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

AbstractIt is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number

Recursive generation of IPR fullerenes

We describe a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm stays entirely within the class of IPR fullerenes, that is: every IPR fullerene is constructed by expanding a smaller IPR fullerene unless it belongs to limited class of irreducible IPR fullerenes that can easily be made separately. The class of irreducible IPR fullerenes consists of 36 fullerenes with up to 112 vertices and 4 infinite families of nanotube fullerenes. Our implementation of this algorithm is faster than other generators for IPR fullerenes and we used it to compute all IPR fullerenes up to 400 vertices.Comment: 19 pages; to appear in Journal of Mathematical Chemistr

A curious family of convex benzenoids and their altans

The altan graph of G, a(G, H), is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2|H|. When G is a polycyclic plane graph with maximum degree 3, the natural choice for the attachment set is to take all perimeter degree-2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a(G, H), as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2. The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a(G, H) over that of G are 2, 1, or 0. Moreover, altans of benzenoids have nullity at least 1. Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1. Here, we exhibit an infinite family of convex benzenoids with 3-fold dihedral symmetry (point group D3h) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid