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.

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.

It 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.

The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent 13 if their symmetric difference forms the edge set of a hexagon of B. A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B.P contains at least one perfect matching, or if B.P is empty. It is proven that there exists a surjective map f 15 from the set of hypercubes of R(B) onto the resonant sets of B such that a k-dimensional hypercube is mapped into a resonant set of cardinality k.

Extremal fullerene graphs with the maximum Clar number ∗