Kekulé Structures in Benzenoid Hydrocarbons

This text is an attempt to outline the basic facts concerning Kekul€ structures in benzenoid hydrocarbons: their history, applica­ tions and especially enumeration. We further pOint out the numerous and often quite remarkable connections between this topic and various parts of combinatorics and discrete mathematics. Our book is primarily aimed toward organic and theoretical chemists interested in the enume­ ration of Kekule structures of conjugated hydrocarbons as well as to scientists working in the field of mathematical and computational chemistry. The book may be of some relevance also to mathematicians wishing to learn about contemporary applications of combinatorics, graph theory and other branches of discrete mathematics. In 1985, when we decided to prepare these notes for publication, we expected to be able to give a complete account of all known combi­ natorial formulas for the number of Kekule structures of benzenoid hydrocarbons. This turned out to be a much more difficult task than we initially realized: only in 1986 some 60 new publications appeared dealing with the enumeration of Kekule structures in benzenoids and closely related topics. In any event, we believe that we have collec­ ted and systematized the essential part of the presently existing results. In addition to this we were delighted to see that the topics to·which we have been devoted in the last few years nowadays form a rapidly expanding branch of mathematical chemistry which attracts the attention of a large number of researchers (both chemists and mathematicians).

1 — Introduction.- 1.1 Benzenoid Hydrocarbons.- 1.2 Historical Remarks.- 1.3 Importance of Kekulé Structures in the Theory of Benzenoid Hydrocarbons.- 2 — Benzenoid Systems: Basic Concepts.- 2.1 Introduction.- 2.2 Definitions and Relations.- 2.3 Classifications of Benzenoids.- 3 — Kekulé Structures and Their Numbers: General Results.- 3.1 Introduction.- 3.2 Theorems About K Numbers.- 3.3 Vertices and Edges in Kekulé Structures.- 3.4 Lower and Upper Bounds of K.- 3.5 Benzenoids with Extremal K.- 3.6 Generation of Normal Benzenoids.- 3.7 Isoarithmicity.- 4 — Introduction to the Enumeration of Kekulé Structures.- 4.1 Schematic Survey.- 4.2 Empirical Methods.- 4.3 Combinatorial Formulas, Especially for the Single Linear Chain.- 4.4 Recurrence Relations for Single Linear and Zigzag Chains.- 4.5 Summation Formulas for Single Linear and Zigzag Chains.- 4.6 Algorithms for Single Linear and Zigzag Chains.- 4.7 Combinatorial Formula for the Single Zigzag Chain.- 4.8 Treatment of a Pericondensed Benzenoid: The Parallelogram.- 4.9 General Remarks.- 4.10 Other Methods.- 5 — Non-Kekuléan and Essentially Disconnected Benzenoid Systems.- 5.1 Introduction.- 5.2 Introductory Examples.- 5.3 The Müller-Muller-Rodloff Rule.- 5.4 Characterization of Concealed Non-Kekuléan Benzenoid Systems.- 5.5 Segmentation.- 6 — Catacondensed Benzenoids.- 6.1 Previous Work.- 6.2 Single Unbranched Chain.- 6.3 Branched Chain.- 6.4 Catacondensed Ladder.- 6.5 Catacondensed All-Benzenoids and Related Systems.- 6.6 Limit Values Involving K Numbers.- 7 — Annelated Benzenoids.- 7.1 Definitions.- 7.2 Previous Work.- 7.3 Annelation to a Linear Chain.- 7.4 Annelation to a Zigzag Chain.- 7.5 Further Developments.- 7.6 Discussion of the Formulas.- 7.7 Algorithm.- 7.8 Dictionary of K Numbers withRelevance to Annelation.- 7 9 Annelation of Two Single Chains.- 7.10 Annelations of Special Benzenoids.- 8 — Classes of Basic Benzenoids (I).- 8.1 Introduction.- 8.2 Hexagon.- 8.3 Chevron.- 8.4 Ribbon.- 8.5 Parallelogram.- 9 — Classes of Basic Benzenoids (II): Multiple Zigzag Chain.- 9.1 Definition.- 9.2 Previous Work.- 9.3 Auxiliary Benzenoid Class.- 9.4 Recurrence Relations for A (n,m) with Fixed Values of n.- 9.5 Combinatorial K Formulas for A (n,m,l) With Fixed Values of m.- 9.6 Combinatorial K Formulas for Z (m,n) With Fixed Values of m.- 9.7 The Polynomial Pm(n) = K{Z(m,n)}.- 9.8 Algorithm.- 9.9 Some General Formulations.- 10 — Regular Three-, Four- and Five-Tier Strips.- 10.1 Previous Work.- 10.2 Definitions.- 10.3 Classification of Regular t-Tier Strips.- 10.4 Examples of Non-Regular t-Tier Strips.- 10.5 Dictionary of K Formulas For Regular 3-, 4- and 5-Tier Strips.- 10.6 Methods of Derivation of K Formulas for t-Tier Strips.- 10.7 The 4-Tier Zigzag Chain.- 11 — Classes of Basic Benzenoids (III).- 11.1 Introduction.- 11.2 Pentagons.- 11.3 Triangles.- 11.4 Streamers and Goblets.- 12 — Classes of Basic Benzenoids (IV): Rectangles.- 12.1 Definitions.- 12.2 Prolate Rectangle.- 12.3 Oblate Rectangle.- 12.4 Auxiliary Benzenoid Classes.- 12.5 Modified Oblate Rectangles.- 12.6 Some General Formulations Concerning Oblate Rectangles.- 13 — Regular Six-Tier Strips and Related Systems.- 13.1 Introduction.- 13.2 Six-Tier Strips.- 13.3 Supplement to the Methods of Derivation of K Formulas For t-Tier Strips.- 13.4 Auxiliary Benzenoid Classes.- 13.5 Two-Parameter K Formulas for Some Multiple Chains.- 13.6 Generalized Auxiliary Class.- 13.7 Étagère.- 13.8 Some Seven-Tier Strips: A Summing UP.- 14 — Determinant Formulas.- 14.1 Introduction.- 14.2 Hexagon.- 14.3Chevron.- 14.4 Ribbon.- 14.5 Parallelogram.- 14.6 Zigzag Chains.- 14.7 Pentagons.- 14.8 Oblate Rectangle.- 15 — Algorithm: A Generalization.- 15.1 Introduction.- 15.2 General Principles.- 15.3 Multiple Chains.- 15.4 Multiple Chains with Truncated Rows.- 15.5 Parallelogram with Truncated and Augmented Rows.- 15.6 Constructable Benzenoids.- 16 — Pericondensed All-Benzenoids and Related Classes.- 16.1 Introductory Remarks.- 16.2 All-Benzenoid Classes Including Modifications.- 16.3 Reticular All-Benzenoids.- 17 — Benzenoids with Repeated Units.- 17.1 Introduction.- 17.2 Fused Repeated Units.- 17.3 Condensed Repeated Units.- 17.4 Benzenoids with Hexagonal and Trigonal Symmetries.- 18 — Distribution of K, and Kekulé Structure Statistics.- 18.1 Introduction and Previous Work.- 18.2 Distribution of K.- 18.3 Average Values of K, and Related Quantities.- 18.4 Number of Normal Benzenoids with a Given K.