The so-called leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentag-onal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spher-ical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerene is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which the leapfrog and other transformations can be accomplished. 1.

Fibonacenes (zig-zag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on their Kekule--structure--related and Clar--structure--related properties.

Many chemical properties of benzenoid hydrocarbons can be understood in terms of the maximum number of mutually resonant hexagons, or Clar number, of the molecules. Hansen and Zheng (1994) formulated this problem as an integer program and conjectured, based on computational results, that solving the linear programming relaxation always yields integral solutions. We prove this conjecture showing that the constraint matrices of these problems are unimodular. This establishes the integrality of the relaxation polyhedra since the linear programs are in standard form. However, the matrices are not, in general, totally unimodular as is often the case with other combinatorial optimization problems that give rise to integral polyhedra. Similar results are proved for the Fries number, another optimization problem for benzenoids. In a previous paper, Hansen and Zheng (1992) showed that a certain minimum weight cut cover problem defined for benzenoids yields an upper bound for the Clar number and...

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A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar number is no more than ⌊n−12 6 ⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60 and C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−12 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.

A simple classically based “mean field ” resonance-theoretic approach is described to anticipate the effects of various modes of electron pairing (as in π-bond formation) for general “alternant” benzenoid conjugated π-networks. This simplified approach avoids generation and manipulation of individual resonance structures, whence application is facilitated, even for very large systems – so large that there might be hundreds or millions or moles or even substantially greater numbers of resonance structures. Some of the predictions so facilitated for several general circumstances are conveniently manifested as explicit algebraic formulas for numbers of unpaired electrons. The simplicity (and apparent reliability) of these algebraic formulas is emphasized; they involve no more than counts of different “kinds ” of π-center carbons, e.g., the numbers of π-centers of different functionalities (primary, secondary, or tertiary) which are “starred ” (in an “alternant ” system) as well as those which are “unstarred”. The argumentation also provides some information about rough locations for the (unpaired) spin densities, and ultimately also about the presence of low-lying excited states, magnetic moments, reactivities, and more. In particular these easily used ideas, which are extensions of those already familiar in “classical ” organic chemistry should then aid in nano-technological developments.

We show that the sextet pattern count of every fullerene is strictly smaller than the Kekulé structure count. This proves a conjecture of Zhang and He [J. Math. Chem. 38(3):2005, p. 315–324]. 1

It is shown that the number of Clar formulas of a Kekuléan benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to the Cl(B)-dimensional hypercube, where Cl(B) is the Clar number of B.

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Extremal fullerene graphs with the maximum Clar number ∗