Description:
Most, yet not all, chemical substances consist of molecules. The fact that molecules have a 'structure' is known since the middle of the 19th century. Since then, one of the principal goals of chemistry is to establish the relationships between the chemical and physical properties of substance and the structure of the corresponding molecules. Countless results along these lines have been obtained along these lines and presented in different publications in this field. One group uses so-called topological indices. About 20 years ago, there were dozens of topological indices, but only a few with noteworthy chemical applications. Over
time, their numbers have increased enormously. At this moment here is no theory that could serve as a reliable guide for solving this problem. This book is aimed at giving a reasonable comprehensive survey of the present, fin de siècle, state of art theory and practice of topological indices.
Contents:
Preface; Chapter 1: Introduction to Molecular Topology: 1.1 Graphs, 1.2 Walks, 1.3
Chemical Graphs; Chapter 2: Topological Matrices: 2.1 Adjacency Matrix, 2.2 Laplacian Matrix, 2.3 Distance Matrix, 2.4 Detour Matrix, 2.5 3D-Distance Matrices, 2.6 Combinatorial Matrices, 2.7 Wiener Matrices, 2.8 Szeged Matrices, 2.9 Path Matrix, 2.10 Hosoya Matrix, 2.11 Cluj Matrices, 2.12 Distance-Extended Matrices, 2.13 Reciprocal Matrices, 2.14 Walk Matrices, 2.15 Schultz Matrices, 2.16 Layer and Sequence Matrices; Chapter 3: Polynomials in Chemical Graph Theory: 3.1 Introduction, 3.2 The Characteristic Polynomial, Part 1, 3.2 The Matching Polynomial, 3.4 The Characteristic Polynomial, Part 2, 3.5 A Unifying Approach: the mð Polynomial, 3.6 The Laplacian Polynomial, 3.7 Moving in Another Direction: the Independence Polynomial, 3.8 More; Chapter 4: Topological Indices, 4.1 Indices Based on Adjancency Matrix, 4.2 Indices Based on Wiener, Distance and Detour Matrices, 4.3 Indices Based on Reciprocal Matrices, 4.4 Indices Based on Combinations of Matrices, 4.5 Indices based on Polynomial Coefficients, 4.6 Indices Based on Eigenvalues and Eigenvectors, 4.7 Indices Based on Graphical Bond Orders, 4.8 Indices Based on the Layer Matrices, 4.9 Indices Based on the Information Theory, 4.10 Other Topological Indices; Chapter 5: Szeged Indices: 5.1 Introduction, 5.2 Definition of the Szeged Index, 5.3 Further Relations between Szeged and Wiener Indices, 5.4 Methods for the Calculation of
the Szeged Index, 5.5 Extensions: Szeged Matrices, Hyper-Szeged and Harary-Szeged Indices, 5.6 Chemical Applications of the Szeged Index; Chapter 6: Cluj Indices: 6.1 Cluj Indices, CJ and CF, 6.2 Cluj Indices in particular Graphs, 6.3 Distance Extended Cluj-Type
Preface:
Many, yet not all, chemical substances consist of molecules. The fact that molecules have a "structure" is known since the middle of the XIX century. Since then, one of the principal goals of chemistry is to establish (causal) relations between the chemical and physical (experimentally observable and measurable) properties of substance and the structure of the corresponding molecules. Countless results along these lines have been obtained, and their presentation comprise significant parts of textbooks of organic, inorganic and physical chemistry, not to mention treatises on theoretical chemistry.
The vast majority of such "chemical rules" are qualitative in nature. A trivial example: if the molecule possesses a -COOH group then the corresponding chemical compound (usually, but not always) exhibits an acidic behavior.
A century-long tendency in chemistry is to go a step further and to find quantitative relations of the same kind. Here, however, one encounters a major problem.
Molecular structure (to simplify: the features expressed by means of structural formulas) is a non-numerical notion. The measured physico-chemical properties of substances are quantities that are expressed by numbers (plus units, plus experimental errors). Hence, to find a relation between molecular structure and any physico-chemical property, one must somehow transform the information contained in the molecular structure into a number (or, more generally, into a sequence of numbers). Nobody knows how to make this transformation or these transformations.
At this moment there is no theory that could serve as a reliable guide for solving this problem. There have been many many many attempts in this direction. One group of them uses so-called topological indices. A topological index is a quantity that is somehow calculated from the molecular graph and for which we believe (or, sometimes, are able to demonstrate) that it reflects relevant structural features of the underlying molecule.
This book is aimed at giving a reasonably comprehensive survey of the present, fin de siècle, state-of-the-art of the theory and practice of topological indices. Some twenty years ago there were a dozen or so topological indices, only few of them with noteworthy chemical applications. Nowadays, their number increased enormously. The readers of this book are warned that in Chapter 7 the number of distinct topological indices will exceed 10,000. An alternative title of our book could be "Topological Indices - A Jungle Guide". There are two nasty, but inevitable questions: Is there any need for topological indices? Is there any real benefit for chemistry (or to generalize: for mankind) from the usage of topological indices?
Some twenty years ago these authors would certainly offer "yes" as answers, but would have a hard time to convince the less gullible part of the chemical community.
Nowadays, the answers are still "yes", but their justification is much easier. The applications of topological indices reached a level when they are directly used for designing pharmacologically valuable compounds. Let the titles of some recently published papers speak for themselves: Quantitative Structure-Activity Relationship Studies on Local Anesthetics [S.P. Gupta, Chem. Rev. 1991, 91, 1109-1119]; Structure-Activity Study of Antiviral 5-Vinylpyrimidine Nucleoside Analogs Using Wiener's Topological Index [S. Mendiratta, A. K. Madan, J. Chem. Inf. Comput. Sci. 1994, 34, 867-871]; Structure-Activity Study on Antiulcer Agents Using Wiener's Topological Index and Molecular Topological Index [A. Goel & A. K. Madan, J. Chem. Inf. Comput. Sci. 1995, 35, 504-509]; Modelling Antileukemic Activity of Carboquinones with Electrotopological State and Chi Indices [J. D. Gough & L. H. Hall, J. Chem. Inf. Comput. Sci. 1999, 39, 356-361]. Of all recent successes made by the aid of topological indices we mention just one. The paper G. Grassy, B. Calas, A. Yasri, R. Lahana, J. Woo, S. Iyer, M. Kaczorek, R. Floc'h, & R.Buelow, Computer Assisted Rational Design of Immunosuppressive Compounds, [Nature Biotechnol. 1998, 16, 748-752] reports on a search for peptides possessing immunosuppressive activity. They used 27 structuredescriptors, of which 12 topological indices. From a combinatorial library of about 280,000 compounds they selected 26 peptides for which high activity was predicted.
Five of them were actually synthesized and tested experimentally. The most potent of these showed an immunosuppressive activity approximately 100 times higher than the lead compound.
One may suspect that in pharmaceutical companies many analogous researches have been (and are currently being) undertaken, with even better results, but - understandably - are not publicized.
* * *
Returning to topological indices: They, of course, are not the miraculous philosopher's stone of our times. They are far from other powerful tools of theoretical chemistry (such as thermodynamics or quantum mechanics). They, however, offer a meager hope to connect structure with properties, and to do this in a quantitative manner.
They, perhaps, deserve the attention of a limited group of chemists. They, perhaps, deserve that every chemist should know a bit about them. They, perhaps, deserve to be mentioned in (undergraduate) courses of organic, physical and pharmacological chemistry.
* * *
Although each author contributed to the entire book, Chapters 1, 2, 4, 6 and 8 were written by M.V.D., Chapters 3 and 5 by I.G. and Chapters 7 and 9 by L.J. Each author takes responsibility only for the materials outlined in the chapters written by himself.
Cluj and Kragujevac, Fall 1999