1 - Introduction.- 1.1 Benzenoid Hydrocarbons.- 1.2 Historical Remarks.- 1.3 Importance of Kekule 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 Kekule Structures in Molecular Orbital Theory.- 1.3.7 Kekule 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 Kekule 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 - Kekule Structures and Their Numbers: General Results.- 3.1 Introduction.- 3.2 Theorems About K Numbers.- 3.3 Vertices and Edges in Kekule 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 Kekule 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 Huckel Molecular Orbitals.- 4.10.3 Different Combinatorial Methods.- 4.10.4 Conjugated Circuits and Kekule 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-Kekulean and Essentially Disconnected Benzenoid Systems.- 5.1 Introduction.- 5.2 Introductory Examples.- 5.2.1 Concealed Non-Kekulean Benzenoids.- 5.2.2 Essentially Disconnected Benzenoids.- 5.3 The Muller-Muller-Rodloff Rule.- 5.4 Characterization of Concealed Non-Kekulean 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 of Regular t-Tier Strips.- 10.3.1 Series of t-Tier Strips.- 10.3.2 Types of Straight Strips.- 10.3.3 Multiple Chain.- 10.3.4 General Classification.- 10.3.5 Special Classification up to t=5.- 10.4 Examples of Non-Regular t-Tier Strips.- 10.5 Dictionary of K Formulas For Regular 3-, 4- and 5-Tier Strips.- 10.5.1 General Features.- 10.5.2 Tables.- 10.6 Methods of Derivation of K Formulas for t-Tier Strips.- 10.6.1 Schematic Survey.- 10.6.2 Application of More General Combinatorial Formulas.- 10.6.3 Stripping.- 10.6.4 Chopping.- 10.6.5 Addition of Algorithm Numerals.- 10.7 The 4-Tier Zigzag Chain.- 11 - Classes of Basic Benzenoids (III).- 11.1 Introduction.- 11.2 Pentagons.- 11.2.1 Definitions.- 11.2.2 Prolate Pentagon.- 11.2.3 Problate Pentagon.- 11.2.4 Oblate Pentagon.- 11.2.5 Mirror-Symmetrical Seven-Tier Pentagons.- 11.2.6 Auxiliary Benzenoid Classes.- 11.3 Triangles.- 11.3.1 Definitions.- 11.3.2 Previous Work and General Formulas.- 11.3.3 Triangles Without Apex.- 11.4 Streamers and Goblets.- 11.4.1 General.- 11.4.2 Streamer.- 11.4.3 Goblet.- 12 - Classes of Basic Benzenoids (IV): Rectangles.- 12.1 Definitions.- 12.2 Prolate Rectangle.- 12.3 Oblate Rectangle.- 12.3.1 Previous Work.- 12.3.2 Compilation of Some Results.- 12.3.3 Limit Values Involving K Numbers.- 12.4 Auxiliary Benzenoid Classes.- 12.5 Modified Oblate Rectangles.- 12.5.1 Introduction.- 12.5.2 Modifications of Rj(m,l).- 12.5.3 Modifications of Rj(m,2).- 12.5.4 Modifications of Rj(m,3).- 12.5.5 Modifications of Rj(m,4).- 12.5.6 Modifications of Rj(m,5).- 12.6 Some General Formulations Concerning Oblate Rectangles.- 12.6.1 Summation K Formulas in Terms of Auxiliary Benzenoids.- 12.6.2 Matrix Formulation.- 12.6.3 Further Developments.- 13 - Regular Six-Tier Strips and Related Systems.- 13.1 Introduction.- 13.2 Six-Tier Strips.- 13.2.1 Classification.- 13.2.2 Dictionary of K Formulas for Regular 6-Tier Strips.- 13.3 Supplement to the Methods of Derivation of K Formulas For t-Tier Strips.- 13.3.1 Introduction.- 13.3.2 First Example.- 13.3.3 Second Example.- 13.4 Auxiliary Benzenoid Classes.- 13.5 Two-Parameter K Formulas for Some Multiple Chains.- 13.6 Generalized Auxiliary Class.- 13.7 Etagere.- 13.8 Some Seven-Tier Strips: A Summing UP.- 14 - Determinant Formulas.- 14.1 Introduction.- 14.2 Hexagon.- 14.3 Chevron.- 14.4 Ribbon.- 14.5 Parallelogram.- 14.5.1 Parallelogram L(m,n).- 14.5.2 Parallelogram Augmented with a Row, L(n,m,l).- 14.6 Zigzag Chains.- 14.6.1 Single Zigzag Chain.- 14.6.2 Double Zigzag Chain.- 14.6.3 Multiple Zigzag Chain.- 14.6.4 Multiple Zigzag Chain Augmented with a Row.- 14.7 Pentagons.- 14.7.1 Prolate Pentagon.- 14.7.2 Problate Pentagon.- 14.7.3 Oblate Pentagon.- 14.7.4 Problate or Oblate Pentagon Augmented with a Row.- 14.8 Oblate Rectangle.- 14.8.1 Oblate Rectangle Rj(m,n).- 14.8.2 Incomplete Oblate Rectangles and Their Associated Class, B(n, 2m-2, l) and B(n, 2m-2, -l).- 14.8.3 Further Developments.- 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.- 15.5.1 Introduction.- 15.5.2 Generalization.- 15.5.3 Algorithmic Rules.- 15.6 Constructable Benzenoids.- 16 - Pericondensed All-Benzenoids and Related Classes.- 16.1 Introductory Remarks.- 16.2 All-Benzenoid Classes Including Modifications.- 16.2.1 Introduction.- 16.2.2 Annelated Pyrenes.- 16.2.3 Pericondensed All-Benzenoid Ladder.- 16.3 Reticular All-Benzenoids.- 17 - Benzenoids with Repeated Units.- 17.1 Introduction.- 17.1.1 Previous Work and Definitions.- 17.1.2 Introductory Examples.- 17.2 Fused Repeated Units.- 17.2.1 General Formulation.- 17.2.2 Special Formulas.- 17.2.3 Formulas when U = L(k,m).- 17.2.4 Formulas for Fused and Annelated Pynenes.- 17.3 Condensed Repeated Units.- 17.3.1 Compressed Parallelograms.- 17.3.2 Compressed Rhombs.- 17.3.3 Compressed Rhombs without Corners.- 17.3.4 General Formulation.- 17.4 Benzenoids with Hexagonal and Trigonal Symmetries.- 17.4.1 Hexagonal Symmetry.- 17.4.2 Trigonal Symmetry.- 18 - Distribution of K, and Kekule Structure Statistics.- 18.1 Introduction and Previous Work.- 18.2 Distribution of K.- 18.2.1 Ranges of K.- 18.2.2 Distribution of K for Normal Benzenoids.- 18.2.3 Distribution of K for Essentially Disconnected Benzenoids.- 18.3 Average Values of K, and Related Quantities.- 18.4 Number of Normal Benzenoids with a Given K.- 18.4.1 Generation of Normal Benzenoids.- 18.4.2 Theoretical Solution.- 18.4.3 Practical Solution: Sieve Method.- 18.4.4 Listing of Results.- 18.4.5 The Fading-Out Phenomenon.