This text is an attempt to outline the basic facts concerning KekulEUR 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.