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Indacenoids are polygonal systems consisting of two pentagons each and arbitrary numbers of hexagons. They represent a class of polycyclic conjugated hydrocarbons. A fixed bond of a polygonal system is an edge which is or is not selected in all the Kekule structures of that polygonal system. It is proved that each catacondensed indacenoid has some fixed single bonds, but has no fixed double bonds. Moreover, the location of fixed single bonds in a catacondensed indacenoid is identified. Finally, a theorem about fixed single bonds in pericondensed indacenoids is proved.

A coronoid subgraph H = {P,ei,i=0,1,…,g-1} is given: P is a six-sided polyhex (benzenoid hydrocarbon) with holes, and ei is an external edge of P; in the outer and inner boundaries of P one side is parallel to the opposite side. H is then cut into slices (subgraphs of H). A set of rules (Lemmas) for such slices is presented; each rule gives an order to the numbers of single and/or double bonds in a slice of H. The Lemmas thus determine whether or not each site has an essential single bond in the outer boundary of P.

The benzenoid chemical formulas, C n H s are treated. Extremal benzenoids and some of their subclasses are defined, and some general mathematical formulations are reviewed, especially those of relevance to the constant-isomer series. Substantial extensions of the existing data on the numbers of benzenoid C n H s isomers are reported.

In families of benzenoid isomers, the molecules with maximum (minimum) Kekule structure counts possess maximum (minimum) stability and other distinguishable properties. These extremal isomers are determined in the case of symmetric di-fibonaceno-annelated benzenoid species B ( m , n ), obtained from a benzenoid system B by attaching to its two symmetric sites fibonacene fragments of length m and n , where m + n = constant.

Polygonal systems Pq, which consist solely of q-gons, are considered. A Pq system corresponds to a polycyclic conjugated hydrocarbon with a (chemical) formula CnHS. Relations between seven selected invariants of Pq are given. The circumscribing of Pq is explained, and the invariants of k-fold circumscribed Pq are determined. The solutions are essentially different for q = 6 and q > 6. In conclusion, two open problems concerning Pq systems are formulated.

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.

A previous conjecture about the building-up of fully-naphthalenoids is proved and formulated as a theorem. the theorem is used to specify a simple procedure which decides whether or not a given benzenoid is fully-naphthalenoid.

An alkanoid is defined as a tree (in the graph-theoretical sense) which can be embedded in the diamond lattice. It corresponds to a staggered conformer of an alkane. A computer program for enumerating non-overlapping alkanoids is described. In these systems there is no pair of coalescing vertices. Results for the unbranched and branched alkanoids are given separately, and the symmetry groups of the systems are determined.

A fully-naphthalenoid system (or simply fully-naphthalenoid) is a benzenoid where all vertices can be covered by disjoint naphthalene units. Invariants of these systems are studied with special emphasis on the CnHs formulae. An extensive enumeration of fully-naphthalenoids by means of a computer program is reported. A detailed classification of the numbers includes the symmetry distribution and an enumeration of the CnHs isomers. The forms of the smallest fully-naphthalenoids are displayed together with their Kekule structure counts.

The fundamental building-up principle states that three types of addition of hexagons are sufficient for generating all benzenoids. The principle is applied to benzenoid CnHs isomers with the result of another principle, which implies the three addition units C4H2, C3H and C2.

A fully-naphthalenoid hydrocarbon (FUN) is a benzenoid hydrocarbon whose all π-electrons can be (formally) grouped into disjoint naphthalene-units. The cyclic conjugation in FUNs is studied by means of the energy-effects of their various cycles. It is found that the dominant conjugation modes in FUNs are those involving the 6- and 10-membered cycles of the “full” naphthalene fragments. The cycles belonging to the “empty” domains of a FUN have significantly smaller energy-effects.

The determinant of the adjacentcy matrix, the algebraic structure count and the Kekulé structure count of circulenes are shown to conform to simple expressions containing Fibonacci numbers.

Polygonal systems correspond to polycyclic hydrocarbons. The numbers of C n H s isomers for these systems are studied systematically. Complete mathematical solutions are reported for systems with the numbers of polygons (or rings), r , up to five, and arbitrary polygon (ring) sizes q ⪖ 3. Explicit combinatorial formulae are given for all cases (nine subclasses) with r ⪕ 4. For r ⪕ 5 (23 subclasses) the solutions are presented in terms of generating functions. The number of degrees of freedom, ϕ, depends on the number of independently varying polygon sizes and sites of annelation. This parameter (ϕ), together with the symmetry properties, are used as a novel basis of classification of the polygonal systems.

Completely condensed polycyclic conjugated hydrocarbons are studied with respect to their numbers of C n H s isomers ( # I ) for arbitrary ring sizes. Direct combinatorial methods are applied and based on the ring-edge sum ( Σq ), viz. the sum of all edges in the q -membered rings taken individually. Explicit expressions of # I are reported for two rings ( r = 2) and three rings ( r = 3). The latter case ( r = 3) splits into structures without any internal carbon ( n i = 0) and those with one internal carbon ( n i = 1). Finally, the corresponding problem for r = 4, n i = 1 is solved by means of computer programming.

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.

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.

Benzenoid (chemical) isomers are, in a strict sense, the benzenoid systems compatible with a formula CnHs. Several invariants, including the Dias parameter, are treated and relations between them are given. Many of the relations involve upper and lower bounds. The periodic table for benzenoid hydrocarbons is revisited and new aspects of it are pointed out. In this connection some new classes of benzenoids are defined: extreme-left, protrusive and circular. Extensive tables of enumeration data for benzenoid isomers are presented. Some of their forms are displayed in figures.

Polygonal systems correspond to polycyclic hydrocarbons. The numbers of C n H s isomers for these systems are studied systematically. Complete mathematical solutions are reported for systems with the numbers of polygons (or rings), r , up to five, and arbitrary polygon (ring) sizes q ⪖ 3. Explicit combinatorial formulae are given for all cases (nine subclasses) with r ⪕ 4. For r ⪕ 5 (23 subclasses) the solutions are presented in terms of generating functions. The number of degrees of freedom, ϕ, depends on the number of independently varying polygon sizes and sites of annelation. This parameter (ϕ), together with the symmetry properties, are used as a novel basis of classification of the polygonal systems.

Polyenoid systems (or polyenoids) are trees which can be embedded in a hexagonal lattice and represent CnHn+2 polyene hydrocarbons. Complete mathematical solutions in terms of summations and in terms of a generating function are deduced for the numbers of polyenoids when overlapping edges and/or vertices are allowed. Geometrically planar polyenoids (without overlapping vertices) are enumerated by computer programming. Thus the numbers of geometrically nonplanar polyenoids become accessible. Some of their numbers are confirmed by combinatorial constructions, a pen-and-paper method.

The benzenoid chemical formulas, C n H s are treated. Extremal benzenoids and some of their subclasses are defined, and some general mathematical formulations are reviewed, especially those of relevance to the constant-isomer series. Substantial extensions of the existing data on the numbers of benzenoid C n H s isomers are reported.

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