The limits of applicability of topological indices

Journal of Molecular Structure (Theo&em), 185 (1989) 187-201 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

THE LIMITS OF APPLICABILITY

187

OF TOPOLOGICAL INDICES

D.H. ROUVRAY Department of Chemistry, University of Georgia, Athens, GA 30602 (U.S.A.) (Received 29 June 1988)

ABSTRACT In recent years topological indices have come prominently to the fore on several different fronts. The prime reason for the wide publicity accorded these indices has been the remarkable ability they possess to correlate and predict the properties of a vast range of molecular species. However, although the indices are now seemingly ubiquitous in the chemical context, they are subject to a number of constraints on their realm of applicability. Here we explore these constraints and attempt to draw realistic boundaries within which the indices may be legitimately used. It is noted in passing that current uses of the indices far from exhaust their manifold future possibilities. Mention is made of some of the numerous promising areas of application still awaiting development.

INTRODUCTION

The remarkable abilities of topological indices (TIs) in modeling molecular behavior have propelled them into the limelight in recent years. A seemingly endless succession of published correlations has documented the effectiveness of these indices in predicting the properties of a vast spectrum of chemical substances. Correlations with properties ranging from the physical to the biological have been reported, and in all the excitement it appeared at times as though hardly any molecular property still remained to be correlated with at least one TI. Several review articles [l-5] have examined the various applications of the indices, and three monographs [6-81 have been devoted solely to discussion of them. Yet, in spite of their apparent ubiquity, there can be little doubt that there are important limits to the applicability of TIs. In this paper we set ourselves the task of exploring these limits and of assessing how they may affect the predictive capacity of the indices. Our ultimate objective is a delineation of the boundaries within which TIs may be legitimately used. Topological indices have been in existence and have a history that is far longer than most chemists realize. The first TIs were introduced almost 150 years ago [ 91, and the fact that the same indices are still in regular use today is testament to their great durability and versatility. In his work on the additive behavior of the members of homologous and other series, Kopp [9] first dem-

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onstrated the utility of indices in correlating a number of physico-chemical properties. His approach was simple; he chose well-defined hydrocarbons for his studies and relatively straightforward properties, such as the specific volume and the density. The index he employed was the number of carbon atoms in the hydrocarbon molecule, an index commonly referred to nowadays as the carbon-number index. Since its inception, this particular index has proved its value to many different types of chemists and has led to its more-or-less continual usage down to the present time (see, for example, ref. 10). This index is well known [ 11,121 to provide an effective measure of the molecular volume of a species; for the members of homologous series, the molecular volume is known to be directly proportional to the carbon-number index [ 131. The first topological index that attempted to characterize branched species was put forward by Wiener [ 141 in 1947. A fully satisfactory characterization of the branching in molecular species has not yet been achieved and is likely to remain problematic for some time [ 151. Wiener’s solution was to define an index equal to the sum of the chemical bonds existing between all the pairs of carbon atoms in the molecule under consideration. It is interesting to note that Wiener made use of the carbon skeleton of the molecule only, and thereby initiated a practice which is still in operation today. The carbon skeleton of a molecule is represented by what is nowadays referred to as the hydrogen-suppressed graph of the molecule. The neglect of the hydrogen atoms appears to be justified for two principal reasons: (i) these atoms are small and so add very little to the overall size of the molecule, and (ii) hydrogen atoms are in no way structure-determining in hydrocarbon species, though they can be allowed for if this is deemed necessary. The role of the hydrogen atom in this context will be addressed more fully in our later discussion of molecular size. It is known [ 16 ] that the Wiener index can be expressed as one half the sum of the entries in the distance matrix of the hydrogen-suppressed graph of the molecule. The index has many different applications; these have been described elsewhere [ 171 by the present author. Following Wiener’s pioneering work, some time elapsed before other topological indices were proposed. The next was suggested by Hosoya [ 161, who advocated using a TI based on the number of ways in which disconnected edges could be imbedded in the hydrogen-suppressed graph of a molecule. Hosoya [ 161 is also responsible for introducing the term topological index, which he originally applied to his index but which, by extension, has now come to embrace all the other indices as well. In 1975 Rand% [ 181 published what was to prove to be the most successful TI to date in terms of its spectrum of application. This index, known today as the molecular connectivity index, has since been developed [ 191 into a series of indices. All these indices are based on summations of weighted subgraphs of the hydrogen-suppressed graph of the molecule. If the subgraphs are vertices, the indices are said to be of zero order, edges yield indices of first order; and so on. Two entire monographs [ 6,8] have been devoted to discussion of these indices and their numerous applications.

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After Randic’s early work, TIs began to appear at a fairly rapid rate; far too rapid in fact for us to examine all of them here. To date, almost 120 different indices (including information-theoretical indices) have been put forward. Readers interested in more information on the indices themselves should consult any of the several reviews and monographs available on the subject [l-8]. THE SIGNIFICANCE

OF TOPOLOGICAL INDICES

As will be apparent from the introduction, topological indices are actually graph invariants. By this is meant that they characterize the hydrogen-suppressed graph of a molecule in terms of a mathematical expression that is independent of both the orientation of the graph and the order in which its vertices may be numbered. In the case of TIs, the expression assumes the form of a numerical scalar descriptor. Figure 1 illustrates the derivation of such a descriptor (the carbon-number index) for an alkane molecule. It should be mentioned in passing that a variety of other invariants have also been employed for the characterization of molecular species, examples being codes [ 201, ma-

Molecule I Chemical Formulo

7

H

H

H

H-+-;-H HA

l!i H-C-H

il Structural Formula I

Chemical Graph I n=5 Topological Index

Fig. 1. Schematic representation of the derivation of a TI (the carbon-number index) from an alkane molecule.

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trices [21], and polynomials [22], although we shall not be concerned with these here. To possess a graph at all, a molecule must comprise a fixed number of atoms held together in a well-defined configuration by relatively strong chemical bonds. Thus, chemical species which are poorly defined or transient, such as intermediates, hydrogen-bonded species, or species that interact strongly with their environment to form charge-transfer complexes, are not suitable for characterization by TIs. In general, therefore, the molecules considered need to be well-defined and distinct entities that can be viewed as existing in isolation from their environment. The idea of describing molecules by means of mathematical invariants has an interesting history and it is instructive to recall the salient features here. In 1861 Couper [ 231 had first advocated the use of the straight-line link to symbolize the very complex set of electronic interactions associated with the formation of a chemical bond. The idea was elaborated by Crum Brown [ 241 who proposed the use of the structural formula to represent chemical species. This development was of crucial importance, for Crum Brown maintained that the structural formula portrayed the relative physical positions of the atoms in space. The structural formula thus depicted the actual linkages that existed between the atoms in a molecule and provided a representation of what is nowadays referred to as the topology or connectivity of the molecule [25]. The mathematician Cayley first recognized that such formulas were, in essence, structures now known as chemical graphs [ 261. It was the mathematician Sylvester who took this process one stage further by representing chemical graphs in terms of graph invariants [ 271. This fundamental idea has since been exploited by numerous workers in the field of structure-activity relationships. For example, Bandit [ 281 has emphasized that the invariants should be treated in the same way as a measured property of a molecule as both arise from and are characteristic of the individual structure in question. From this vantage point, graph invariants may be seen to bridge the seemingly disparate concepts of molecular structure and chemical properties in that they provide a numerical description of structure. The question as to what the invariants actually represent at the molecular level is very interesting and one that has not been satisfactorily resolved at the present time. It is evident that TIs could, in theory, reflect a large number of different molecular characteristics, which for convenience we shall classify as either geometric or non-geometric. Examples of the two types of characteristics are listed in Table 1 together with key references for those seeking more detailed information. The geometric characteristics we list (molecular surface area, cross-sectional areas, volume, conformation) can all be approximated by calculations based on a few experimentally determined parameters. The parameters that need to be known are the van der Waals radii of the atoms involved, the interatom bond distances, and the bond angles. However, such de-

191 TABLE 1 A list of some of the molecular characteristics that can be modeled with topological indices Parameter modeled

Ref.

Geometric Conformation Cross-section Surface area Volume

8,65,66 67,68 8,14,36,68 8,30-32

Non-geometric Branching Chirality Complexity Cyclicity Flexibility Shape Similarity Symmetry

8,29,69 25 70 71 8 72,73 74,75 76,77

terminations are always made with a fair degree of uncertainty because of the indeterminate positions of the electrons in atoms and the arbitrariness associated with calculating atomic overlaps. The non-geometric characteristics cannot be estimated in an absolute sense at all, as any calculation of them has to depend ultimately on an arbitrary definition of what is being measured [ 291. Because independent estimates can be made of the geometric characteristics of molecules, it is possible to assess how well different indices correlate with such characteristics. To illustrate the types of correlation that have been obtained, we shall now consider the parameters molecular volume and surface area. Although these have been investigated by a number of workers, we focus here on the results obtained by Motoc and Balaban [30], Charton [31], and Kier and Hall [32]. Motoc and Balaban [30], showed that TIs varied enormously in their ability to reflect the van der Waals volume of molecules. Whereas the Wiener and Randic molecular connectivity indices yield very high correlations (r> 0.99),the Hosoya index gives only a moderate correlation (r= 0.88),and a number of other TIs have even smaller correlation coefficients. These workers concluded that the marked differences in the correlations were due to “compositional inhomogeneity” of the TIs and that the best suited TI for a given correlation would therefore need to be determined by trial and error. Charton [31] demonstrated that the molar volume of a molecule calculated in different ways [ 33-351 also correlated very well with the van der Waals volume, and that some relationships between steric-effect parameters and volume parameters was inevitable. Kier and Hall [ 321 found that counts

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of the a; II, and lone-pair electrons in atoms correlated very well (r = 0.99) with van der Waals volumes. In all cases the volumes of the hydrogen atoms play a relatively minor role and may be neglected to a first approximation. Kier [ 361 also showed that the surface area of a molecule correlates closely (~0.99) with the Randic’ molecular connectivity index. ESTABLISHING CORRELATIONS WITH TOPOLOGICAL INDICES

Repeated reference has been made above to correlations involving TIs and, in particular, the correlation coefficient, r, has been mentioned several times. It should be evident that TIs can be used effectively only if reliable correlations can be established between them and a variety of molecular properties and parameters. It is important to emphasize, however, that there is no need of statistics for calculating the indices themselves. TIs can be calculated with 100% accuracy by following the prescriptions of the relevant algorithm, which represents a considerable advantage whenever TIs are used to substitute for experimental properties that are either difficult to measure or inaccessible. The need for statistics arises only after the TIs have been determined, and it then becomes of crucial importance to ensure that valid statistical techniques are adopted. This issue has proven to be a rather vexatious one, and various workers have pointed out [ 37-391 that many published correlations are in fact less significant than they were originally reported to be. Topliss and co-workers [37,40] have discussed the possibility of chance factors yielding spuriously good correlations in multiple-regression analyses involving TIs and other parameters. They constructed plots showing the number of measured data points that would be needed to maintain the probability of encountering a chance correlation at no greater than 1%. Thus, if a total of 10 different TIs were to be screened for possible correlation with a given property, 20 data points would be required to hold the chance correlation level at a maximum of 1%. This same issue was also addressed by Klopman and Kalos [ 411 from a slightly different standpoint. These workers observed that coincidental correlations could arise from both a statistical bias build into the data and chance factors, and that such occurrences were found when the TIs were incorporated in a stepwise regression analysis and also when tried out successively. Moreover, they demonstrated that correlation equations of the general type P=k+

5 CiTi i=l

where P is some observed property that is correlated with m topological indices, Tip and k and ci are constants having little or no statistical significance if m is large owing to the relatively high number of degrees of freedom. Klop-

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man and Kalos [41] also proposed that the magnitude of the Fisher statistic, F, could be used as a measure of the reliability of a correlation. By assigning cutoff values for F, this parameter effectively weeded out the majority of meaningless correlations. While it is true that statistics can never be used to prove causality but only to establish that a significant correlation exists, there is a certain amount of corroborative evidence that TIs yield generally good correlations because they reflect two of the most important parameters that determine molecular behavior. In attempting to predict the properties of 114 diverse liquids, Cramer [42,43] found that only two types of intermolecular interaction were important as determinants of the observed physical properties, namely the bulk and the bulk-corrected cohesiveness of the molecule in question. Translated into our terminology, these parameters are respectively equivalent to the van der Waals volume of the molecule and a shape-dependent variable, such as a topological-shape descriptor. On the basis of his studies, Cramer [42,43] concluded that any property that depends primarily on non-specific and non-covalent intermolecular interactions can be predicted from the molecular structure of the species involved. He stressed that the correlations he obtained were not artefacts of the factor analysis process he had employed, but that they arose rather from the circumstance that all of the properties he investigated are ultimately determined by a similar intermolecular interaction mechanism. The large number of correlations obtained to date with TIs attest to the ubiquitous operation of this kind of interaction in the natural world. CONSTRAINTS ON THE APPLICABILITY

OF TOPOLOGICAL INDICES

From the foregoing it is apparent that TIs can function as effective descriptors of molecular behavior only if a number of important criteria are satisfied. These criteria inevitably impose certain constraints on the range of applicability of the indices; our objective in this section is to enumerate and explore these constraints. We start by considering the interaction of an individual molecule with its environment which we suppose to be made up of numerous identical molecules. Such a molecule will experience inter- and intra-molecular forces, the cumulative effects of which lead to correlation curves of the general type shown in Fig. 2. In this figure, the boiling points of normal alkane species are plotted against their Wiener index and the plot exhibits a pronounced curve. This type of behavior is typical of most properties of homologous series. Attempts to straighten the curve by plotting it on logarithmic scales results in a much straighter line, as can be seen in Fig. 3. Note, however, that log-log plots are not straight lines because the properties are determined by the three-dimensional conformation of molecules. As the average conformation changes with increasing number of atoms, i.e. the fractal dimensionality of the molecule

194 6.90 r

0.0

Wiener Index

20

4.0

6.0

6.0

10.0

In (Wiener Index 1

Fig. 2. A plot of the boiling point against the Wiener index for the first 40 normal alkanes. Fig. 3. A plot of the boiling point against the Wiener index for the first 40 normal alkanes using logarithmic scales. TABLE 2 A list of the common long range and short range intermolecular forces Intermolecular force

Additivity

Long range Dispersion Besonance Magnetic

Nearly additive Non-additive Very weak

Short range Orbital overlaps Electrostatic Induction

Non-additive Additive Non-additive

changes [ 441, we cannot expect logarithmic plots to exist in the form of straight lines. This means that simple linear relationships between the molecular properties of the members of homologous series and TIs are never observed in practice. Let us now examine a little more closely the inter- and intramolecular forces operating on a single molecule in the bulk phase. The intermolecular forces have been described and catalogued by numerous workers [ 41. These forces are generally classified as either long range or short range [ 461, and are known to be attractive at long range and repulsive at short range. In Table 2 a listing of these various forces is given based on the work of Buckingham [ 471. For alkane molecules, and indeed for the members of most homologous series, the principal intermolecular interactions involve dispersion forces. Instantaneous dipoles generated in a molecule as a result of fluctuations in its electron density induce dipoles in neighboring molecules. The molecules in question experience this type of interaction as a mutually attractive force. In more complex systems

195

the interactions become far more complicated, and are easier to understand if we adopt a simplified model. A reasonable approximation to the description of extramolecular interactions is provided by the point-charge model of the molecule [ 481 in which the nuclei and electrons comprising a molecule are envisioned as a single point charge in space. This model, which is valid only for long-range interactions, enables us to calculate the electric potential existing between a pair of molecules in the bulk phase. It is this potential.which governs the nature of the interaction between the molecules. This intermolecular potential must remain effectively constant, or scale in a regular fashion with increasing molecular size, if molecular topology is to be the decisive factor in determining the intermolecular interactions between molecules. There are similar constraints on the intramolecular forces extant within a molecule that need to be satisfied if molecular topology is to play a predominant role. The theory of atoms in molecules [49] is based on the idea that the volume of space occupied by a molecule can be partitioned into disjoint sets representing mononuclear regions or atoms. Each of the atoms defined in this way is assumed to be associated with some average physico-chemical or other property. Summation of all the average atomic properties yields the average molecular property for the entire molecule. The familiar notion of the functional group which exhibits more-or-less constant properties throughout a series of organic or inorganic reactions derives from the same reasoning. The theory may be viewed as a generalization of quantum-mechanical concepts to certain subsystems of the total molecular system, and, as such, represents a quantum-chemical formulation of the long-established additivity principle [491. This principle, which states that molecular properties of a species may be determined from simple summation of the contributions to the property from each of its constituent parts, was first enunciated in embryonic form by Kopp [501 in 1842. It has found wide application in the prediction of numerous thermodynamic and physico-chemical properties of hydrocarbons and other species [ 51,521. The principle is now seen to rest on the foundation of transferability of the force constants between atoms of equivalent types in a molecule. Such transferability will hold provided that there are no complicating factors present in species, e.g. steric repulsion or ring or bond strain, and in this kind of situation the molecular topology will be the dominant factor in determining molecular behavior. STRATEGIES FOR COPING WITH THE CONSTRAINTS

In the case of alkanes there is a very high degree of bond transferability. A vibrational analysis of these species based on bond transferability led to the calculation of some 350 frequencies with an overall error of no more than 0.25% [ 531. Other homologous series show similar high levels of transferability. Thus, Duke and O’Leary [ 541 have demonstrated that quantum-mechanical calcu-

196

lations on various hydrocarbon species can yield results of ab initio quality on the assumption of bond transferability. Homologous series are, therefore, almost ideal for the application of topological indices. Even substituted species in such series show excellent correlations [ 6,8]. The presence of heteroatoms, i.e. atoms other than carbon or hydrogen, in a species introduce certain problems, although they are not insurmountable. A strictly topological analysis of heteroatom-substituted species would regard every atom in the species as identical. However, in practice a weighting factor must be applied to such atoms to allow for their differing size and electronegativity. The various approaches adopted to date to the weighting of heteroatoms are summarized in Table 3. In general, it must be admitted that no completely satisfactory way of weighting heteroatoms (and thus the vertices and edges of the corresponding graph) has yet been found, although the methods now in use give reasonable correlations

[81. A variety of other complicating factors may be encountered in the application of TIs. A number of these factors are listed in Table 4 and for each appropriate references are given. The presence of multiple bonds in a species can normally be allowed for by making use of graphs that have multiple edges, e.g. a double bond is frequently represented by a graph-theoretical lune. Somewhat greater problems are encountered when the bonds cannot be classified as either single, double or triple, as is the case in the benzene molecule. In such circumstances, the practice that is usually followed is the adoption of a graph that is appropriate to the problem in hand. In benzene, for example, the edges of the graph may be weighted with some factor which may differ from one. Accordingly, therefore, differing graphs may be used for the same species depending on the nature of the problem being investigated. Branching in species is not associated with any special difficulty, though it should be commented that the TABLE 3 A list of the approaches adopted to date to allow for the presence of heteroatoms in molecular species Description of approach

Applicability

Ref.

Empirical weighted graph vertex valences

Molecular connectivity indices

18

Non-empirical weighted graph vertex valences

Molecular connectivity indices

79

Htickel weighted graph edges

Wiener index

80

Information-theoretical

Information-theoretical

partitioning of vertices

indices

81

Atomic-number weighted distance matrix elements

Wiener index

82

Relative electronegativity graph-edge weighting

Balaban J index

83

Covalent radius graph-edge weighting

Balaban J index

83

197 TABLE 4 A list of various complicating factors that are encountered in the derivation of topological indices Complicating factor

Ref.

Branching Conformational changes Cycles (rings) Heteroatoms Hydrogen bonding Multiple bonds Specific receptor interaction Steric effects

8,29,69 44,66 71 78-83 57 698 a4 85

Fig. 4. The scatter plot obtained for the 75 isomeric decanes when the boiling point is plotted against the Wiener index.

correlations obtained for the structural isomers of a given member of a homologous series are, in general, very poor. As an illustration of this, the scattered distribution obtained for the set of decane isomers when the boiling point is plotted against the Wiener index is shown in Fig. 4; the correlation coefficient for this particular set [ 51 is only 0.0035! The presence of rings in species can be accommodated in most instances straightforwardly. One problem that can arise here is that the rings may be possessed of high strain energy [ 551; in such cases new forces are introduced that usually cannot be incorporated in a simple topological analysis. In fact, only if a series of molecules is considered in which the ring strain varies in a consistent way with ring size can TIs yield good correlations. Correlations involving the biological activity of molecules usually involve a number of additional considerations because the mechanisms of biointeractions are in general more subtle than their physical or chemical counterparts. For example, ill-defined bonds may be encountered as is the case in the phenomenon of hydrogen bonding [ 561. Hydrogen bonds are typically about one tenth the strength of a covalent bond, and in water can be thought of as forming flickering random networks of irregular and distorted linkages [ 571. In such a

198

situation our requirement that the species under study be modeled in terms of well-defined chemical graphs is not met. In fact, the best that can be done at present [57] is to define so-called “connectivity pathways” that can be described by computer simulation models. In correlating the bioactivity of substances in different organisms, separate data sets need to be constructed for each organism in order that an appropriate parametrization can be made. Such parametrization is possible only if the same mechanism operates for all the substrates in all the organisms studied. MacFarland’s model of bioactivity [ 581 implies that three principal steps are involved in the manifestation of bioactivity: (i) delivery of the substance from its entry point into the organism via biomembranes to the receptor site; (ii) recognition of the substance by the receptor and the formation of a substrate-receptor complex; and (iii) possible reaction of the substance to yield new chemical products. Any one or any combination of these steps may be decisive in determining the level of bioactivity exhibited by the substance in question. The behavior of the substance during the recognition process at the receptor has been classified by Trinajstic et al. [ 591 into four basic types: additive, constative, multiplicative, and derivative. Whenever the former is dominant, use can be made of TIs to establish correlations with the bioactivity; the action at the receptor is then said to be of nonspecific type [2,60]. Fortunately, a wide range of biological phenomena fall within the domain of the additive. The use of TIs to model these phenomena has been reviewed by various workers [ 2,3,60]. More sophisticated techniques, such as three-dimensional QSAR, are generally required to model the other types of biological interaction [ 591.

SUMMARY AND CONCLUSIONS

Topological indices possess the remarkable ability of being able to correlate and predict a very wide spectrum of properties for a vast range of molecular species. In spite of their apparent ubiquity on many fronts, however, they are subject to a number of important constraints. We have analysed these constraints and have come to the conclusion that two are of overwhelming importance: the inter- and intra-molecular forces experienced by an individual molecule in the bulk phase. In the case of the former, the electric potential existing between pairs of molecules must remain effectively constant or scale in a moreor-less regular fashion with increasing molecular size. In the case of the intramolecular forces, there must exist an approximate transferability of the force constants between atoms of equivalent types within any series of molecules being correlated. Only if these two basic criteria are met, and there is an ab-

199

sence of perturbing factors such as steric repulsion or excessive strain energy in the molecule, will the use of TIs be strictly justified. The members of homologous and congeneric series appear to satisfy these criteria almost perfectly. Biological activity can be modeled by means of TIs if the interactions are of so-called non-specific type, which implies that the behavior of the substrate at the receptor is essentially additive. Fortunately, a wide range of biological interactions appear to fall within this domain. In spite of the severe constraints that TIs are subject to, surprisingly good correlations can sometimes be achieved in situations that seemingly do not match up to these stringent criteria. Thus, in a study on the incipient formation of soot from hydrocarbon fuel molecules [ 611, TIs yielded some remarkable correlations even though the process of soot formation involves the initial fuel molecule in a complex sequence of chemical transformations. Similarly, studies on the carcinogenicity of polycyclic aromatic hydrocarbon molecules [62] have found excellent correlations of carcinogenicity with the topological structure of the hydrocarbon itself, although the latter is known to undergo many transformations [ 631 before the ultimate carcinogen is reached. Perhaps even more surprisingly, a study on the electronic absorption bands in these same hydrocarbons revealed that better correlations could be obtained by using reduced chemical graphs of these species rather than their full graphs [ 641. It would seem that there is still much to learn about the nature and role of TIs; they may well still hold many more surprises in store. AKCNOWLEDGEMENT

We are indebted to the U.S. Office of Naval Research for partial support of this research project.

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58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85

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