Some New Trends in Chemical Graph Theory Ramo ´ n Garcı ´a-Domenech, ², | Jorge Ga ´ lvez, ², ⊥ Jesus V. de Julia ´ n-Ortiz, ‡,# and Lionello Pogliani* ,§ Unidad de Investigacio ´n de Disen ˜ o de Farmacos y Conectividad Molecular, Departamento de Quı ´mica Fisica, Facultad de Farmacı ´a, Universitat de Vale ` ncia, 46100 Burjassot, Vale ` ncia, Spain, Instituto de Tecnologia Quimica, CSIC-Universidad Politecnica de Valencia, Av. de los Naranjos s/n, 46022 Vale ` ncia, Spain, and Dipartimento di Chimica, Universita ` della Calabria, via P. Bucci 14/C, 87036 Rende (CS), Italy Received April 19, 2007 Contents 1. Introduction 1128 1.1. Short Introduction 1128 1.2. Main Graph Concepts 1129 1.3. QSAR/QSPR Model Studies 1131 1.4. Plot Methods 1132 2. New Trends in Molecular Connectivity 1132 2.1. Mathematical Tools 1132 2.1.1. Complete Graphs and the Core Electron Representation 1132 2.1.2. Hydrogen Perturbation 1135 2.1.3. Molecular Connectivity and Pseudoconnectivity Basis Indices and their Duals 1136 2.1.4. Higher-Order Terms 1137 2.1.5. Variable Molecular Connectivity Index 1137 2.2. Modeling with the Complete Graph and the Hydrogen-Perturbed Algorithms 1138 2.2.1. Binding, ∆H L °, and Lattice, ∆H G °, Enthalpies and the Polarizability of Metal Halides 1138 2.2.2 Polarizability of Organic Compounds 1140 2.2.3 Partition Coefficient of Halogenated Organic Compounds 1141 2.2.4 Three Properties of Halomethanes 1143 2.2.5 Boiling Points of Amines and Alcohols 1145 2.3. Model of Properties with the Variable Descriptor 1146 3. From Graph Invariants to Lead Design 1147 3.1. Mathematical Tools 1147 3.1.1. Descriptors 1147 3.1.2. Modeling 1148 3.1.3. Multilinear Regression Analysis (MLRA) 1148 3.1.4. Validation of the Selected Equations 1148 3.1.5. Linear Discriminant Analysis (LDA) 1149 3.1.6. Pharmacological-Activity Distribution Diagrams (PDDs) 1149 3.2. Molecular Selection and Design 1150 3.2.1. Database Search with External Validation 1150 3.2.2. Molecular Selection by Virtual Screening on Databases 1150 3.2.3. Virtual Combinatorial Syntheses and Computational Screening 1151 3.2.4. Molecular Design of New Structures 1151 3.3. Applications 1154 3.3.1. Prediction of Physicochemical Parameters 1155 3.3.2. Prediction of Pharmacological Properties 1155 3.3.3. Mathematical Models for the Selection and Design of New Active Compounds 1155 3.3.4. New Biological Activities Discovered through Virtual Screening and Molecular Design 1155 4. Chemical Kinetics and Chemical Graph Theory 1155 4.1. Kinetic Graphs 1155 4.1.1. Kinetic Mechanisms 1155 4.1.2. Categories and Subcategories 1158 4.1.3. Types 1158 4.1.4. Classes 1159 4.2. Kinetics Graphs and Rate Laws 1159 5. Application of Chemical Graph Theory to Biomacromolecules 1160 5.1. Descriptors for Biomacromolecules 1160 5.1.1. Polypeptides 1160 5.1.2. Polynucleotides 1161 6. Meaning of Basic Molecular Connectivity Indices 1162 6.1. The Molecular Connectivity IndexsA Quantum Interpretation 1162 6.2. The Molecular Connectivity IndexsA Kinetic Interpretation 1162 6.3. The Molecular Connectivity IndexsA Geometric Interpretation 1163 6.4. The Molecular Connectivity IndexsA Variable Index Interpretation 1164 7. Conclusions 1164 8. Acknowledgments 1165 9. Appendix 1165 9.1. An Update 1165 9.2. Graph-Theoretical Software for Model Purposes 1165 9.2.1. APPROBE (& POLLY) 1165 9.2.2. CODESSA 1165 9.2.3. CLUJTOPO 1166 9.2.4. DRAGON 1166 9.2.5. MOLGEN-4.0 1166 9.2.6. MOLCONN-Z 1166 9.2.7. TOPS-MODE and MODesLab 1166 9.2.8. The Variable Connectivity Index 1167 9.2.9. Toolkit 1167 9.3. Mathematical Chemistry Monographs 1167 10. Note Added in Proof 1167 11. References 1167 * To whom correspondence should be addressed. E-mail: lionp@unical.it. ² Universitat de Vale `ncia. ‡ CSIC-Universidad Politecnica de Valencia. § Universita ` della Calabria. | E-mail: ramon.garcia@uv.es. ⊥ E-mail: jorge.galvez@uv.es. # E-mail: jejuor@uv.es. 1127 Chem. Rev. 2008, 108, 1127-1169 10.1021/cr0780006 CCC: $71.00 © 2008 American Chemical Society Published on Web 02/27/2008