G. Clar 6n Rule versus Hückel 4n + 2 Rule 3464 H. Hydrocarbons versus Heteroatomic Systems 3465 IV. Hidden Treasures of Kekulé Valence Structures 3466 A. Conjugated Circuits 3467 B. Innate Degree of Freedom 3470 C. Clar Structures 3472 V. Graph Theoretical Approach to Chemical Structure 3473 A. Metric 3473 B. Chemical Graphs 3473 C. Isospectral Graphs 3473 D. Embedded Graphs 3475 E. Partial Ordering 3476 VI. On Enumeration of Benzenoid Hydrocarbons 3477 VII. Kekulé Valence Structures Count 3479 A. Non-branched Cata-condensed Benzenoids 3481 B. Branched Cata-condensed Benzenoids 3482 C. Benzenoid Lattices 3482 D. Peri-condensed Benzenoids 3483 E. Miscellaneous Benzenoids 3484 F. The Approach of Platt 3485 G. Computer Programs for Calculating K 3485 H. Transfer-Matrix Method 3486 I. Use of Recursion Relations 3486 J. Use of Signed Matrices 3487 VIII. Enumeration of Conjugated Circuits 3488 IX. Approximate Approaches versus Ambitious Computations 3490

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Aromaticity of polycyclic conjugated hydrocarbons.

Graph theory has been used for modeling chemical structures, that is, chemical compounds, intermediates, changes, reactions and mechanisms. Among the most important such applications are: (1) structure-property relationships (SAR); (2) design of compounds of desired properties; (3) enumeration and construction of compounds of certain classes; (4) unique representation of compounds (for documentation needs). Here, we outline selected accomplishments of the past, show some current efforts and point to directions which may lead to further advancements in chemical graph theory.

In search of structural invariants

The relations of caterpillar trees (which are also known as Gutman trees and benzenoid trees) to other mathematical objects such as polyhex graphs, Clar graphs, king polyominos, rook boards and Young diagrams are discussed. Potential uses of such trees in data reduction, computational graph theory, and in the ordering of graphs are considered. Combinatorial and physical properties of benzenoid hydrocarbons can be studied via related caterpillars. It thus becomes possible to study the properties of large graphs such as benzenoid (i.e. polyhex) graphs in terms of much smaller tree graphs. Generation of the cyclic structures of wreath and generalized wreath product groups through the use of caterpillar trees is illustrated.

Applications of caterpillar trees in chemistry and physics

The use of power sum symmetric functions leads to Newton's identities, which relate the traces of various powers ofA, the adjacency matrix of a graph, and the coefficients of the characteristic polynomials. While it is possible to solve Newton's identities and generate the coefficients by recursion or, alternatively, to derive them by sequential manipulations (yielding the explicit formulas), we show how the results can be expressed using a combinatorial approach and relate the evaluation of the coefficients to selected Young diagrams.

On the evaluation of the characteristic polynomial via symmetric function theory

It is shown that the Clar number of a benzenoid hydrocarbon H (defined as the number of circles in a Clar formula, or equivalently as the maximum number of mutually resonant hexagons of H) can be determined by mixed-integer programming. Moreover, linear programming appears to suffice in practice to find in moderate computing time the Clar number of pericondensed hydrocarbons with more than a thousand hexagons.

The Clar Number of a Benzenoid Hydrocarbon and Linear Programming

Clar structures recently used as basis-set to compute resonance energies [9] are identified as maximal independent sets of benzenoid hydrocarbons “colored” in a special way. Binomial properties of such objects are induced for several catafusenes and perifusenes (Eqs. 2–31). Novel polynomials, called Clar polynomials, are given for perifusens in terms of units of catafusenes which allow display and enumeration of the populations of their Clar structures. The work is particularly pertinent to that of [8] and [9].

Combinatorial Clar sextet theory

A recursion exists between the absolute magnitudes of the coefficients of the characteristic polynomials of certain families of cyclic and acyclic graphs which makes their computation quite easy for very large graphs using a pencil-and-a-paper approach. Structural requirements are given for such families of graphs which are of interest to the problem of recognition defined in [1].

Sherif El-Basil

Papers

Combinatorial Clar sextet theory

Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]