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.- 1.3.1 General.- 1.3.2 Total ?-Electron Energy.- 1.3.3 Resonance Energies.- 1.3.4 Pauling Bond Order.- 1.3.5 Miscellaneous Applications.- 1.3.6 Kekulé Structures in Molecular Orbital Theory.- 1.3.7 Kekulé Structures in the Aromatic Sextet Theory.- 2 - Benzenoid Systems: Basic Concepts.- 2.1 Introduction.- 2.2 Definitions and Relations.- 2.2.1 Definition of Benzenoid Systems.- 2.2.2 Helicenic and Coronoid Systems.- 2.2.3 Vertices and Edges.- 2.2.4 Graphs.- 2.2.5 Further Definitions and Relations.- 2.2.6 Coloring of Vertices.- 2.2.7 Perfect Matchings and Kekulé Structures.- 2.2.8 All-Benzenoid Systems.- 2.2.9 Modes of Hexagons.- 2.3 Classifications of Benzenoids.- 2.3.1 First Classification of Benzenoids.- 2.3.2 Second Classification of Benzenoids ("neo").- 2.3.3 Third Classification of Benzenoids (? values).- 2.3.4 Fourth Classification of Benzenoids (Symmetry).- 2.3.5 Results of Enumeration 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.- 4.2.1 Systematic Drawings.- 4.2.2 Method of Fragmentation.- 4.2.3 Degenerate Systems.- 4.2.4 Modified Method of Fragmentation.- 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.- 4.8.1 Introduction.- 4.8.2 Algorithm.- 4.8.3 Auxiliary Benzenoid Class and Its Application.- 4.8.4 The Auxiliary Class and the Algorithm Numerals.- 4.8.5 Recurrence and Summation Formulas.- 4.9 General Remarks.- 4.10 Other Methods.- 4.10.1 Introduction.- 4.10.2 Application of Coefficients in Hückel Molecular Orbitals.- 4.10.3 Different Combinatorial Methods.- 4.10.4 Conjugated Circuits and Kekulé Structures.- 4.10.5 Application of Polynomials.- 4.10.6 Analytical Expressions for the Determinant of the Adjacency Matrix.- 4.10.7 Algorithmic Formula for All-Benzenoids.- 4.10.8 Fully Computerized Method.- 4.10.9 Transfer-Matrix Method.- 4.10.10 Computer Programs.- 5 - Non-Kekuléan and Essentially Disconnected Benzenoid Systems.- 5.1 Introduction.- 5.2 Introductory Examples.- 5.2.1 Concealed Non-Kekuléan Benzenoids.- 5.2.2 Essentially Disconnected Benzenoids.- 5.3 The Müller-Muller-Rodloff Rule.- 5.4 Characterization of Concealed Non-Kekuléan Benzenoid Systems.- 5.4.1 Introduction.- 5.4.2 First Characterization.- 5.4.3 Second Characterization.- 5.5 Segmentation.- 5.5.1 Introduction.- 5.5.2 Tracks and Partial Differences Between the Numbers of Valleys and Peaks.- 5.5.3 Characterization Based on the Numbers t and s.- 6 - Catacondensed Benzenoids.- 6.1 Previous Work.- 6.2 Single Unbranched Chain.- 6.2.1 Introductory Remarks Including some Helicenic Systems.- 6.2.2 Algorithm.- 6.2.3 Single Linear and Zigzag Chains: Combinatorial Formulas.- 6.2.4 Other Single Chains.- 6.3 Branched Chain.- 6.3.1 Systems with One Branching Hexagon.- 6.3.2 Systems with Several Branching Hexagons.- 6.4 Catacondensed Ladder.- 6.4.1 Introduction.- 6.4.2 The Case of m=2.- 6.4.3 General Case.- 6.5 Catacondensed All-Benzenoids and Related Systems.- 6.5.1 Some General Properties and Some Examples.- 6.5.2 Class with Only 2-Segments in the Backbone and Related Classes.- 6.5.3 Class with Only 3-Segments in the Backbone and Related Classes.- 6.5.4 Other 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.- 7.3.1 Introduction.- 7.3.2 One-Sided Annelation.- 7.3.3 Two-Sided Annelation.- 7.4 Annelation to a Zigzag Chain.- 7.5 Further Developments.- 7.5.1 Some Auxiliary Results.- 7.5.2 Further Developments of the Formulas for Two-Sided Annelations.- 7.6 Discussion of the Formulas.- 7.6.1 Even and Odd Systems.- 7.6.2 One-Sided Annelations.- 7.6.3 Two-Sided Annelations.- 7.7 Algorithm.- 7.8 Dictionary of K Numbers with Relevance to Annelation.- 7 9 Annelation of Two Single Chains.- 7.9.1 Introduction.- 7.9.2 Annelation of Two Linear Chains.- 7.9.3 Annelation of Two Chains of Which at Least One is a Zigzag Chain.- 7.9.4 Extended Application of the Algorithm.- 7.10 Annelations of Special Benzenoids.- 7.10.1 Introduction.- 7.10.2 Tabulation of Formulas.- 8 - Classes of Basic Benzenoids (I).- 8.1 Introduction.- 8.2 Hexagon.- 8.2.1 Definition.- 8.2.2 Previous Work and General Formulas.- 8.2.3 Dihedral and Regular Hexagonal Hexagons.- 8.2.4 Limit Values Involving K Numbers.- 8.3 Chevron.- 8.3.1 Definition.- 8.3.2 Previous Work and General Formulas.- 8.3.3 Algorithm.- 8.3.4 Derivation of Formulas.- 8.3.5 Mirror-Symmetrical Chevrons.- 8.3.6 Generalized Chevron.- 8.4 Ribbon.- 8.4.1 Definition.- 8.4.2 Previous Work and General Formulas.- 8.4.3 Derivation of Formulas.- 8.4.4 Algorithm.- 8.4.5 Generalized Ribbon.- 8.5 Parallelogram.- 8.5.1 Definition.- 8.5.2 Previous Work and General Formulas.- 8.5.3 Rhomb.- 8.5.4 Auxiliary Benzenoid Class.- 8.5.5 Algorithm.- 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.- 9.8.1 Multiple Zigzag Chain, A(n,m), and the Class A(n,m,l).- 9.8.2 Truncated Multiple Zigzag Chain.- 9.9 Some General Formulations.- 9.9.1 Summation K Formulas in Terms of Auxiliary Benzenoids.- 9.9.2 Matrix Formulation.- 9.9.3 The Case of n=3.- 9.9.4 Alternative Approach.- 10 - Regular Three-, Four- and Five-Tier Strips.- 10.1 Previous Work.- 10.2 Definitions.- 10.2.1 Regular t-Tier Strip.- 10.2.2 Straight and Skew Strips: The Top-Bottom Shift.- 10.3 Classification …