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In this note a theorem concerning the coincidence between the characteristic polynomial of a cycle and the polynomial of Kekule structure count of a primitive coronoid is presented which implies a complete solution of Hosoya's mystery.

The Kekule structures of benzenoid hydrocarbons of type A(n, m) are enumerated. Here A(n, m) stands for a system composed ofn condensed zigzag aromatic chains, each containingm hexagons. We study the hitherto unresolved case when the chain lengthm is variable and the number of chainsn is constant.

A benzenoid, or a benzenoid system, is a connected planar graph whose every interiorface is a regular hexagon. A peak (resp. valley) of a benzenoid is a vertex which lies above(resp. below) all its first neighbors. A Kekulean benzenoid is a benzenoid with at least oneperfect matching. An essentially disconnected benzenoid is a Kekulean benzenoid which hassome fixed bonds. Essentially disconnected benzenoids have proved to be very useful incertain enumeration techniques for Derfect matching. Hence the problem of recognizing

Benzenoid and coronoid systems are considered. A bond order is defined in terms of the elements of the inverse of a skew-symmetric adjacency matrix. It is conjectured that it is identical with the Pauling bond order. A computer program was designed for computing Pauling bond orders of Kekulean coronoids in general. Numerical examples are given. The skew-symmetric adjacency matrix was exploited for recognition of essentially disconnected coronoid systems. The 29 smallest essentially disconnected coronoids with the phenalene hole are depicted.

The constant-isomer series are studied in relation to the periodic table of benzenoid hydrocarbons. General properties of the table are formulated in terms of explicit expressions. In particular a general expression for the formulas (CnHs) of the ground forms of the constant-isomer series was achieved. The numbers of the isomers (cardinalities of CnHs) are divided into to parts. Explicit equations are given for the first part of the cardinalities.

Simply connected polyhexes with one internal vertex (n; = 1 perifusenes) are enumerated. Complete mathematical solutions are given in terms of summation formulas and by generating functions. The symmetry is accounted for. The numerical results are used to derive numbers of simply connected helicenic (geometrically nonplanar) polyhexes withn i = 1.

A general formula for the Kekule structure count (K) is deduced for the class of catacondensed all-benzenoids with unbranched backbones. The formula is extended to “thin” pericondensed all-benzenoids, where allowance is made for pyrene units. In this treatment the fragmentation matrices are employed. A generalization of these matrices is furnished. Next some generalK formulas for classes of catacondensed and thin pericondensed all-coronoids are deduced. Again the fragmentation matrices are employed, but the problem is also studied in terms of certain polynomials.

Hollow hexagons form a subclass of primitive coronoid systems. The macrocyclic aromatic hydrocarbon kekulene corresponds to a hollow hexagon. The hollow hexagons for given numbers of hexagonal units (h) were enumerated by computer aid, but also an analytical solution for the numbers of hollow hexagons was achieved. For the Kekule structure counts (k) of a hollow hexagon a general formula is reported. Also the maximum and minimumK numbers are considered.

The Kekule structure counts (K) for primitive coronoids are treated. TheK formula which involves the trace of a matrix product is recalled and supplemented with new findings. In this way a kind of symmetry in the mathematical formulations is restored. Another general formulation for theK number is provided in terms of polynomials which, for a somewhat mysterious reason, are identified as the matching polynomials of cycles.

The Kekule structure count (K) of a planar graph (in the graph-theoretical sense) is obtainable as the Pfaffian of a skew-symmetric adjacency matrix. This principle was used to design a general computer program for theK numbers of coronoid systems. Thus it became possible for the first time to determineK, as a matter of routine, during the computer-aided generation and enumeration of all types of Kekulean coronoids, including those where perimeters of [4k] cycles are present. The methods are exemplified by a complete account (with figures andK numbers] of the 27 half essentially disconnected coronoids with the phenalene hole and 11 or 12 hexagons.

A type of graphs derived from a cycle and associated with primitive coronoids are referred to as “crowns”. The characteristic polynomials and matching polynomials of crowns are studied. These notions are used to calculate the sextet polynomial for primitive coronoids. Patterns of aromatic sextets are treated in some detail.

A group-theoretical treatment of theKekule structures of polyphenes is performed. General expressions in terms of the number of benzene rings are given for the symmetrical structure (ΓKek) and the characters (χKek) of the representation on the basis of theKekule structures.

The number ofKekule structures is derived for an arbitrary single-chain aromatic. This class of molecules consists of condensed aromatics where the line through the centers of the fused benzene rings is a single (unbranched) line.

Classes of coronoids (or degenerate coronoids), r and q are obtained by providing the rectangle-shaped benzenoids Rj (m, 3) with a naphthalenic or pyrenic hole, respectively. The numbers of Kekule structures (K) are studied. It was found for theK numbers of the classes in question:r=4/5R andq=1/5R. The classes r′ and q′ are similar to r and q, respectively, but the naphthalenic or pyrenic hole is oriented in a different way. For these classes it was found:r′=3/5R,q′=2/5R.

Recurrence relations and explicit combinatorial expressions are derived for the number ofKekule structures of certain multiple-chain condensed aromatics.

Primitive coronoids of hexagonal symmetry (D6h orC6h) are referred to as “waffles”. Some theorems about unbranched catacondensed benzenoids are presented and used to derive a general combinatorial formula for the number of Kekule structures (K) for waffles. The symmetry-adapted method of fragmentation is employed. SeveralK formulas for special classes of waffles are also reported.

A class of aromatic molecules with fused benzene rings is considered. It is characterized by one, two or three straight chains of benzene rings meeting at one ring. The group-theoretical problem of symmetry in terms of ΓKek χKek is solved; explicit formulas are given for the symmetry groupsD3h andC2v.

A normal coordinate analysis is performed for cyclohexene (C6H10). The calculated frequencies for C6H10 and C6D10 are reported. The mean amplitudes of vibration for all types of interatomic distances in these two molecules were also computed. Those of the bonded distances and nonbonded C−C distances in C6H10 are compared with the corresponding data from election diffraction.