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Chemical Graph Theory-Facts and Fiction Indian Journal of Chemistry Vol. 42A, June 2003, pp. 1207- 1218 Review Chemical graph theory-Facts and fiction Milan Randic National Institute of Chemistry, Hajdrihova 19, Ljubljana. Slovenia Received 31 January 2003 Graph Theory (GT) and its applications in chemistry, the so-call ed Chemical Graph Theory (CGT), appear to be two of the most misunderstood areas of theoretical chemi stry. We outline briefly possible causes for mi sunderstanding and suggest remedies, incl uding a test on the knowledge of GT and CGT. Introduction "primitive." The Conjugated Circuits method Graph Theory (GT) is a not so young branch of enumerates circuits within individual Kekule valence discrete mathematics. It is generally accepted that it structures of polycyclic conjugated hydrocarbons started with Leonhard Euler's paper I on the seven circuits in which there is a regular alternation of CC l9 bridges of Konigsberg published in 1736. It has single and CC double bonds . The outcomes of such received due attention after the first book on Graph enumeration are analytical expressions for molecular 20 Theori, which appeared two hundred years later, was resonance energy (RE). Schaad and Hess have published in 1936. Since then GT became one of the shown that the method of Conjugated Circuits is fastest expanding branched of mathematics, the closely related to Herndon's Resonance Theory21, a importance of which has been particularly recognized variant of VB calculations based solely on the set of in its role with development of the algorithms for Kekule valence structures of a molecule, that has been 22 computer applications of GT3. Graph theory has been in fact considered some time ago by Simpson , but 4 accepted and appreciated in physics as well as in was mostly (undeservingly) overlooked. Let us also biologi, but its acceptance in chemistry has been point out that although the Resonance Theory and the marred with numerous unwarranted obstructions, Conjugated Circuit Model if based on the same despite that it made contributions in chemical parameterization become mathematically equivalent, 6 documentation , structural chemistri, physical the two approaches are conceptually and chemistrl, inorganic chemistr/, quantum computationally different. This has become more 23 chemistry 10, orgamc chemistry II , chemical apparent as defJlOn strated by Klein and coworkers l2 synthesis , polymer chemistry 13, medicinal with application of the Conjugated Circuit model , 4 l5 particularly, to computation of stabilities of chemistri , genomics and DNA studies , and of 24 recent date proteomics 16. fullerenes , for which no other computations offered insights into their stabilities. This paper on the facts and fiction surrounding Chemical Graph Theory should be viewed not only Chemical Graph Theory has been motivated by as equal to other branches of theoretical chemistry but comments received for one recent graph theoretical also as complementary and necessary for better paperl7 in which new molecular descriptors have been understanding of "the nature of the chemical proposed. In some areas of chemistry, notably structure". It is true that at one time it was not physical chemistry, theoretical chemistry and uncommon to see GT misidentified with HMO, the medicinal chemistry, there appears to be continuing Ruckel Molecular Orbitals model of early Quantum hesitation to accept graph theoretical concepts and Chemistry. The likely reason for this is because for methodology as a legitimate theoretical tool. This is graphs of conjugated hydrocarbons the adjacency not the place to list numerous cases of matrix corresponds to the Huckel matrix of HMO mi sunderstanding of CTG but let us mention one such method. In thi s context one could understand that l8 case which reflects the position clearly. In an article Chemical Graph theory was of lesser interest to many on quantum chemical computation of the stability of chemists - but even at that time GT was applied to [n]phenalenes graph theoretical method of numerous diverse problems of chemistry, some listed "Conjugated Circuits" has been referred as in Table 1, which had nothing to do with HMO. 1208 INDIAN J CHEM, SEC A, JUNE 2003 Table I- Areas of appl ication of Topological Indices Physico-chemical properties of molecules, including those having heteroatoms Bi ological activity of drugs, including toxicity Search for pharmacophore Search of large databases Molecular Similarity Molecular di versity Enumeration of isomers Degenerate rearrangements Drug design Screening of combin atorial libraries Characterization of folded proteins Characteri zation of DNA primary sequences Characteri zation of t-RNA Characteri zation of molecular shape Characteri zation of molecular chirality Docking for molecular recognition Numeri cal characterization of proteomi s maps Topological Indices have found use in chemical much different. Graphs, however, have the additional and biological applications and that Chemical graph advantage in that they allow some flexibility in theory has made important conceptual and associating with individual edges and individual quantitative contributions to chemistry. In the later vertices various weights, whi ch can be different in part of this contribution we will list a number of different applications. Embedded graphs are defined important recent contributions of topological indices as graphs of fixed geometry, which may but need not and Chemical Graph Theory to chemistry and let coincide with the geometry of a chemical structure. readers to delineate facts from fiction . Part of the One of the major uses of graphs is to serve as problem may be in that some concepts of GT are so sources of various structural invariants, which is close to the chemical language of structural chemistry tantamount to sayin g various mathematical properties with whi ch many chemists are familiar. Thus many of structure. Thus, compounds that typicall y exhibi t . chemists may get an impression that GT is (if not various physico-chemical properties and biological simple, and even simpli stic) not very sophisticated, acti vities can now tn addition exhibit various and hence not capable of offering proper insights on mathematical properties. There is an important chemical structure. That such a position is false and distinction between the collection of physico­ that GT and CGT are rich in content and include chemical properties and biological activities of a numerous profound propositions can be easily found molecule and the coll ection of its mathematical if one is interested in this subject. properties: The number of physico-chemical properties and biological activities of a molecule is Topological indices as molecular descriptors finite, while in contrast mathematical properties We will only briefly outline the substance of appear unlimited in their number. However, for a topological indices and molecular descriptors in order , mathematical property to be of interest in chemistry it to facilitate readers unfamiliar with details of has to show its use. This may be in structure-property Chemical Graph Theory to form their own view on regressIOns, structure-acti vity relationshi ps, the nature of topological indices and to be able to establishing molecular similarity and diversity, form an opinion on their potential use. For more screentng combinatorial libraries, clustering of detail. readers should consult several of avai lable chemical compounds, design of novel drugs, 25 review articles on topological indices , including also characterization of DNA structures, characterization a brief introduction avai lable in the Encyclopedia of of proteomics maps, etc. (see Table I). G Computational Chemistr/ • Topological indices are The most common use of mathematical invariants, structural invariants based on modeling of chemical which are also known as graph theoretical indices or structures by molecular graphs. Hence, covalent topological indices, is as molecular descriptors in bonds are represented by edges and atoms as vertices. QSPR and QSAR (quantitative structure-property Superficially molecular graphs and molecular relationships and quantitative structure-actIvIty structural formulas that chemists often use are not relationships, respectively). There are at least two RANDle: CHEMICAL GRAPH THEORY 1209 aspects of QSPR and QSAR with slightly different pathways, in agreement with Wong's coevolution emphasis: (1) One is interested in as good as possible theory of the genetic code, were obtained." 30 predictions of properties without being concerned Katritzky, Lobanov & Karelson .3 1 have developed with the interpretation of the descriptors used, if they computer software, which is distributed freely to do the job; (2) One is interested in as good as possible people in academic institutions. CODESSA computes characterization of properties and one is much some 400 molecular descriptors of which about a concerned with the interpretation of the descriptors third are topological indices. This software has been used. In the first case typically one selects a subset of used in numerous applications in QSPR and QSAR structures for testing the model, in the second case ever since it was introduced. one is using all available data and trying to Agrafiotis32 describes a novel diversity metric for "understand" the model. Since properties are usually use in the design of combinatorial chemistry and expressed numerically, clearly if one considers high-throughput screening experiments. We cite a regressions one needs descriptors that will · al
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