Indian Journal of Chemistry Vol. 42A, June 2003, pp. 1207- 1218

Review

Chemical graph theory-Facts and fiction

Milan Randic

National Institute of Chemistry, Hajdri hova 19, Ljubljana. Slovenia

Received 31 January 2003

Graph Theory (GT) and its applications in chemistry, the so-called Chemical Graph Theory (CGT), appear to be two of the most misunderstood areas of theoretical chemistry. We outline briefly possible causes for mi sunderstanding and suggest remedies, incl uding a test on the knowledge of GT and CGT.

Introduction Graph Theory (GT) is a not so young branch of

discrete mathematics. It is generally accepted that it started with Leonhard Euler's paper I on the seven bridges of Konigsberg published in 1736. It has received due attention after the first book on Graph Theori, which appeared two hundred years later, was published in 1936. Since then GT became one of the fastest expanding branched of mathematics, the importance of which has been particularly recognized in its role with development of the algorithms for computer applications of GT3. Graph theory has been accepted and appreciated in physics4 as well as in biologi, but its acceptance in chemistry has been marred with numerous unwarranted obstructions, despite that it made contributions in chemical documentation6

, structural chemistri, physical chemistrl, inorganic chemistr/, quantum chemistry 10, orgamc chemistry II , chemical synthesis l2, polymer chemistry 13, medicinal chemistri4

, genomics and DNA studies l5 , and of recent date proteomics 16.

This paper on the facts and fiction surrounding Chemical Graph Theory has been motivated by comments received for one recent graph theoretical paperl7 in which new molecular descriptors have been proposed. In some areas of chemistry, notably physical chemistry, theoretical chemistry and medicinal chemistry, there appears to be continuing hesitation to accept graph theoretical concepts and methodology as a legitimate theoretical tool. Thi s is not the place to list numerous cases of misunderstanding of CTG but let us mention one such case which reflects the position clearly. In an article l8

on quantum chemical computation of the stability of [n]phenalenes graph theoretical method of "Conjugated Circuits" has been referred as

"primitive." The Conjugated Circuits method enumerates circuits within individual Kekule valence structures of polycyclic conjugated hydrocarbons circuits in which there is a regular alternation of CC single and CC double bonds l9. The outcomes of such enumeration are analytical expressions for molecular resonance energy (RE). Schaad and Hess20 have shown that the method of Conjugated Circuits is closely related to Herndon 's Resonance Theory21, a variant of VB calculations based solely on the set of Kekule valence structures of a molecule, that has been in fact considered some time ago by Simpson22

, but was mostly (undeservingly) overlooked. Let us also point out that although the Resonance Theory and the Conjugated Circuit Model if based on the same parameterization become mathematically equivalent, the two approaches are conceptually and computationally different. This has become more apparent as defJlOnstrated by Klein and coworkers23

with application of the Conjugated Circuit model , particularly , to computation of stabilities of fullerenes24 , for which no other computations offered insights into their stabilities.

Chemical Graph Theory should be viewed not only as equal to other branches of theoretical chemistry but also as complementary and necessary for better understanding of "the nature of the chemical structure". It is true that at one time it was not uncommon to see GT misidentified with HMO, the Ruckel Molecular Orbitals model of early Quantum Chemistry. The likely reason for this is because for graphs of conjugated hydrocarbons the adjacency matrix corresponds to the Huckel matrix of HMO method. In thi s context one could understand that Chemical Graph theory was of lesser interest to many chemists - but even at that time GT was applied to numerous diverse problems of chemistry, some listed 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 Topologica l Indices

Physico-chemical properties of molec ules, including those having heteroatoms Biological activity of drugs, including toxicity Searc h for pharmacophore Search o f large databases Molecular Similarity Molecular di versity Enumeration of isomers Degenerate rearrangements Drug design Screening of combinatorial libraries Characterization of folded proteins Characteri zation of DNA primary sequences Characteri zation of t-RNA Characteri zation of molecular shape Characteri zation of mo lecular chirality Docking for molecular recognition Numerical characteriza tion of proteomi s maps

Topological Indices have found use in chemical and biological applications and that Chemical graph theory has made important conceptual and quantitative contributions to chemistry. In the later part of this contribution we will list a number of important recent contributions of topological indices and Chemical Graph Theory to chemistry and let readers to delineate facts from fiction . Part of the problem may be in that some concepts of GT are so close to the chemical language of structural chemistry with which many chemists are familiar. Thus many chemists may get an impression that GT is (if not simple, and even simpli stic) not very sophisticated, and hence not capable of offering proper insights on chemical structure. That such a position is fa lse and that GT and CGT are rich in content and include numerous profound propositions can be easily found if one is interested in this subject.

Topological indices as molecular descriptors We will only briefly outline the substance of

topological indices and molecular descriptors in order to facilitate readers unfamiliar with details of Chemical Graph Theory to fo rm their own view on the nature of topological indices and to be able to form an opinion on their potential use. For more detail. readers should consul t several of avai lable review articles on topological indices25

, including also a brief introduction avai lable in the Encyclopedia of Computational Chemistr/G

• Topological indices are structural invariants based on modeling of chemical structures by molecular graphs. Hence, covalent bonds are represented by edges and atoms as vertices. Superficially molecular graphs and molecular structural formulas that chemists often use are not

much different. Graphs, however, have the additional advantage in that they allow some flexibility in associating with individual edges and individual vertices various weights, which can be different in different applications. Embedded graphs are defined as graphs of fixed geometry, which may but need not coincide with the geometry of a chemical structure.

One of the major uses of graphs is to serve as sources of various structural invariants, which is tantamount to saying various mathematical properties of structure. Thus, compounds that typically exh ibi t . various physico-chemical properties and biological acti vities can now tn addition exhibit various mathematical properties. There is an important distinction between the collection of physico­chemical properties and biological activities of a molecule and the collection of its mathematical properties: The number of physico-chemical properties and biological activities of a molecule is finite, while in contrast mathematical properties appear unlimited in their number. However, for a

, mathematical property to be of interest in chemistry it has to show its use. This may be in structure-property regressIOns, structure-acti vity relationshi ps, establishing molecular similarity and diversity , screentng combinatorial libraries, clustering of chemical compounds, design of novel drugs, characterization of DNA structures, characterization of proteomics maps, etc. (see Table I) .

The most common use of mathematical invariants, which are also known as graph theoretical indices or topological indices, is as molecular descriptors in QSPR and QSAR (quantitative structure-property relationships and quantitative structure-actIvIty relationships, respectively) . There are at least two

RANDle: CHEMICAL GRAPH THEORY 1209

aspects of QSPR and QSAR with slightly different emphasis: (1) One is interested in as good as possible predictions of properties without being concerned with the interpretation of the descriptors used, if they do the job; (2) One is interested in as good as possible characterization of properties and one is much concerned with the interpretation of the descriptors used. In the first case typically one selects a subset of structures for testing the model, in the second case one is using all available data and trying to "understand" the model. Since properties are usually expressed numerically, clearly if one considers regressions one needs descriptors that will · also numerically characterize a structure - hence a need for topological indices and other structural invariants.

Recent accomplishments in use of topological indices as molecular descriptors

We collected a dozen illustrations from the literature in which topological indices have been used for diverse problems of chemistrl7

.39

, to examine the papers and then judge if topological indices are useful.

Flower27 in his paper on DISSIM, a computer program which addresses the problem of selecting diverse subsets from larger collections of chemical compounds considered relationships of 159 topological indices (listed in his Table 1) of which 39 were used in his CACS (Computer-Aided Compound Selection) protocol.

Lahana and coworkers28 by selecting four from two dozen topological indices for screening a combinatorial library of some 280,000 virtual decapeptides were able, by selecting "windows" of allowed values for individual topological indices and nine other molecular descriptors, to reduce this enormous library to 26 compounds on which they performed more advanced calculations. Finally they selected five compounds and synthesized them finding that one compound had an immunosuppressive activity that was almost hundred times more active than the lead compound!

Andrade and coworkers29 used novel topological indices to characterize t-RNA. They proposed "weighted structural descriptors-closely related to Randic connectivity and Balaban distance indices-as distinctive characteristics of each structure. Molecules were characterized by a set of weighted structural descriptors and classified by a clustering method and discriminant function analysis. Two main groups of tRNAs that correspond to the biosynthetic amino acid

pathways, in agreement with Wong's coevolution theory of the genetic code, were obtained."

Katritzky, Lobanov & Karelson30.3 1 have developed

computer software, which is distributed freely to people in academic institutions. CODESSA computes some 400 molecular descriptors of which about a third are topological indices. This software has been used in numerous applications in QSPR and QSAR ever since it was introduced.

Agrafiotis32 describes a novel diversity metric for use in the design of combinatorial chemistry and high-throughput screening experiments. We cite a short extract of this paper: "The data set used in this study is based on the reductive amination reaction ... and is utilized for the construction of structurally diverse druglike molecules with useful pharmacological properties . . . For demonstration purposes, 300 primary and secondary amines and 300 aldehydes were selected at random . . . and were used to generate a virtual library of 90,000 products . . . Each compound in the 90,000-membered library was characterized by an established set of 117 topological descriptors, which were subsequently normalized and decorrelated using principal component analysis, resulting in an orthogonal set of 23 latent variables which accounted for 99% of the total variance of data

Liu and collaborators33 reported on a molecular electronegativity distance vector based on 13 atomic types, called MEDV -13, as a descriptor for predicting the biological activities of molecules based on QSAR. Their conclusion was that the study on steroids shows that the performance of the MEDV-13 method has comparability with the previous methods containing 3D QSAR, and the peptide study gives a high quality of the QSAR model based on the MEDV -13 method. However, the MEDV -13 method only employs information about an element atom type, valence electron state, and chemical bond type from the 20 molecular topology and requires no information related to 3D structures or physicochemical properties or molecular alignment. So, the MEDV -13 descriptor is fast, easy to use, reproducible and predictable one for the QSAR studies.

Galvez and collaborators34 tried to combine parameters of semiempirical (quantum chemical) calculations with topological indices in order to arri ve at any discriminant function for antibacterial activity. They found that two quantum chemical descriptors (QDs) played important role for discriminant function but at the same time they have shown that these two

1210 INDIAN J CHEM, SEC A, JUNE 2003

QDs can be well expressed by topological indices (TIs). The utility of expressing QDs "as a function of the TIs is to the extent that their predictive capability is for larger sets of compounds for which the calculation of TIs is much faster and easier than the calculation of QOs .. . ". They concluded: "Using the multilinear regression and the LDA (linear discriminant analysis), a pattern of topological simjlarity of antibacterial activity has been obtained. This pattern has been applied successfully for the search of drugs that together with other pharmacological activities can also show antibacterial activity. An added advantage is that this screening can be carried out using large databases and with low time-consuming."

Ga035 reported that the genetic algorithm was very efficient method for variable selection or optimization. Besides available topological indices he also used BCUT metric (an extension of Burden's parameters, which are based on a combination of the atomic number for each atom and a description of the nominal bond type for adjacent and non-adjacent atoms and incorporates both connectivity information and atomic properties). He made binary QSAR analysis of Carbonic Anhydrase II Inhibitors. "The best binary QSAR model was obtained with a combination of 23 molecular descriptors including f . . . d (2 0 v 1 v d 2 V) our connectIvIty 111 exes X, X, X, an X, three shape indexes CZK, 1Ka, 3Ka), log P(o/w), and 15 BCUTs .. . The cross-validated accuracy is 91 % on active compounds, 92% on inactive compounds, and 91 % for all compounds. Thus the predictive power of the binary QSAR model is quite high." He also made binary QSAR analysis of Estrogen Receptor Ligands and obtained sirrular results: 'The best binary QSAR model was obtained with a combination of 24 molecular descriptors including four connectivity indexes (0 X, 1 X, 2 X, and 1 X V), two shape indexes (IKa, 2Ka), flexibility index <1>, log P(o/w), and 16 BCUTs ... The cross-validated accuracy is 73% on active

compounds, 90% on inactive compounds, and 88% for all compounds. "

Yaffe and Cohen36 reported on estimation of vapor pressure using topological indices as descriptors. Here is an extract from their Abstract: A neural network based quantitative structure-property relationship (QSPR) was developed for the vapor pressure­temperature behavior of hydrocarbons based on a data set for 274 compounds. The optimal QSPR model was developed based on 7-29-1 back-propagation neural network architecture using valence molecular

connectivity indices (I XV, 3 xv. 4 XV), molecular weight, and temperature as an input parameters. . . .The performance of the QSPR for temperature-dependent vapor pressure, which was developed from a simple set of molecular descriptors, displayed accuracy of better than or well within the range of other avrulable estimation methods.

Estrada and Molina37 considered the classification of antibacterial activity of 2-furylethylene derivatives. According to them: Topographic (3D) molecular connectivity indices based on molecular graphs weighted with quantum chemical parameters are used in QSPR and QSAR studies. These descriptors were compared to 2D connectivity indices (vertex and edge ones) and to quantum chemical descriptors in modeling partItIon coefficient (log P) and antibacterial activity of 2-furylethylene derivatives. In describing log P the 3D connectivity indices produced a significant improvement (more than 29%) in the predictive capacity of the model compared to those derived with topological and quantum cherrucal descriptors. The best linear discrimjnant model for classifying antibacterial activity of these compounds was also obtained with the use of 3D connectivity indices. The global percent of good classification obtained with 3D and 20 connectivity indices as well as quantum chemical descriptors were 94.1,91.2 and 88.2 respectively.

We included this paper as it clearly shows that graph theoretical approaches are not lirruted to 20 objects (graphs) but can be extended to 3D objects (embedded graphs) . To quote from the introduction of the paper of Estrada & Molina: An important step forward in the development of graph-based molecular descriptors has been the definition of topographic descriptors. These kinds of molecular descriptors are based on molecular graphs with appropriate weights to account for 3D molecular features. The pioneering works in this direction were done by Milan Randic at the end of 1980s. In these works Randic proposed the use of topographic distance matrices, first based on graph embedded on a hexagonal lattice and then on a 3D diamond lattice.

Burden38 considered a large Benzodiazepine Data Set (245 compounds), Muscarine Data Set (162 compounds) and Toxicity Data Set (277substituted benzenes). Here is an extract from his paper: Gaussian processes method (GPM) constitutes a method of solving regression problems. The usual coefficients or weights associated with other regression methods are absent and an exact Bayesian analysis is

RANDle: CHEMICAL GRAPH THEORY 1211

accomplished using matrix manipulations. . . A combination of five set of easily computed indices were employed in this work: the well studied Randic index (R); the valence modification to the Randic index by Kier and Hall (K); and an atomistic (A) index developed by Burden which has now been enhanced by recognition of aromatic atoms and hydrogen atom donors and acceptors (B) . . . Two further indices have been added the first counts the number of rings of various sizes (G) and the second counts some common functional groups (F) .. . The four type of index, R, K, A, and B, are complementary, and we have shown in previous studies that their combination yields better QSAR models than the individual indices alone.

Now if all this is not sufficient to convince readers that topological indices are useful in chemistry and found applications in quite diverse problems then I would like to draw attention of readers to a paper of mine written in collaboration with Novic and Vracko l6

• In this paper the authors explore the characterization of 2-D electrophoresis proteomics maps by certain structural invariants derived from matrices constructed by considering for all pair of spots in a proteomics maps the shortest (Euclidean) distances and distances me red along zigzag lines connecting protein spots of the neighboring abundance. This paper is a sequel to previous papers in which we outlined the idea of characterizing 2-D proteomics maps by graph-theoretical descriptors. To illustrate the approach we selected data of Anderson et al.39 on protein abundance in mouse liver under series of dose of peroxisome proliferator L Y 1711883. We found strong linear correlation between the experimentally applied doses and the leading eigenvalue of Distance/Distance type matrix40

constructed for the experimental proteomics maps.

Discussion Readers will have to judge whether the utility of

topological indices has been demonstrated "beyond the reasonable doubt." Observe that half a dozen "cases" have been taken from not only a single journal, but a single volume of that journal. Moreover, in the same volume of J chem In! Comput Sci. that we screened 28 papers41 are published from the Second Indo-US Workshop on Mathematical Chemistry, held in Duluth, MN, in May 2000 which is "crowded" with papers on application of GT to Chemistry, including numerous publication on novel topological indices. For example, there are papers on: criteria for

classification biological actIvItIes from models of structural similarity; comparison of a neural net-based QSAR algorithm with holograms and multiple linear regression - based QSAR; graph theoretical analysis of tunneling electron transfer in large polycyclic aromatic hydrocarbon networks; on distance related indexes; an extension of the Wiener index and the "overall Wiener index;" interpretation of well-known topological indices, and structural interpretation of several distance-related topological indices ; characterization of DNA primary sequences based on average distances between bases, and characterization of DNA by triplet of nucleic acid bases; illustration of CODESSA-based theoretical QSPR model with variable · molecular descriptors based on distance related matrices; novel shape descriptors for molecular graphs and use of graph shells as molecular descriptors; characterization of 2D chirality; and several papers on the use of the variable connectivity index and the hierarchical approach to QSAR.

Real world chemistry-what is it? In the past some cntIcs of CGT were

"complaining" that researchers in GT are mostly preoccupied with hydrocarbons . Hydrocarbons are also part of the "real-world chemistry". If not why not? In Table 2 are listed a few hydrocarbons and their properties just to remind readers that hydrocarbon chemistry, including fullerenes that may or may not be decorated with hydrogens, may be as fascinating as any branch of chemistry. Fullerenes are even a better illustration of simple chemical graphs in view that usually hydrogens are not present and need not be suppressed!

What critics overlook in objecting to modeling CGT on hydrocarbons is to see that this is only the first stage in developing novel molecular descriptors and models. If a new topological index is not useful in characterizing selected properties of hydrocarbons, they are probably even less suitable for characterizing of properties of heteroatomic compounds. Once descriptors are found useful they, as a rule, are generalized to extend to molecules having heteroatoms. The list of the dozen papers outlined in the previous section has numerous such illustrations.

Concluding remarks We will end with a brief outline of the manuscript

that was the reason behind this review. The manuscript considers several novel topological indices for octane isomers (but equally applicable to other molecules) ,

1212

Formula

CJ2HIO

C 1sHI8

C I9H32

C I9H40

C22H I4

C 21 HI6

C2) H46

C30HSO

C30H62

C.jQHsl

C4oHs6

C4oH68

Ci soHI86

INDIAN J CHEM, SEC A, JUNE 2003

Table 2-Diverse properties of different hydrocarbons

Name Property

heat transfer agent; fungistat for oranges

Guaiazulene anti-inflammatory; anti-ulcerative

Tridecylbenzene detergent; forms stable foam in the presence of fat

Pristane lubricant, ant i-corrosion agent

Pentacene large crystal semiconductor

3-Methylcholanthrene experimental use in cancer research

Muscalure sex pheromone

Squalene oil, agreeable odor; bactericide

Squalane lubricant; transformer oil; perfume fixative; skin lubricant

Lycopene carotenoid occurring in ripe fruit

a-Carotene vitamin A precursor

Phytofluene polyene hydrocarbon widespread in the vegetable kingdom

Hexabenzocoronene derivative liquid crystal (component for photovoltaic films)

all derived from a new graphical matrix. Matrix elements of a graphical matrix, in contrast to the traditional graph theoretical matrices (such as the adjacency matrix, distance, matrix, detour matrix, Wiener and Hosoya matrices, etc.) are not numerical , but are defined through qualified subgraphs of a graph considered. Graphical matrices have been for the first time introduced only a few years ago, and besides the seminal paper42, the paper we are talking about is the only other manuscript considered this subject 17

• Graphical matrices are not widely known even among chemical graph theory, being published in less available and less read mathematical journals. So this particular paper was the first opportunity for a wider circle of readers interested in structural chemistry to hear about graphical matrices!

Table 3-The regression coefficient for quadratic regressions of

Because matrix elements of graphical matrix are non-numerical the first step in applications is to arrive at a numerical representation for such matrices, which is based on a selection of a particular graph invariant to serve as "transition" to a numerical matrix. Once a numerical matrix is constructed, one has choice to select any of already known procedures to extract structural invariants from such a matrix . Thus if one chooses the Wiener index of subgraphs appearing in the matrix as a recipe to arrive at a numerical matrix, and then again choosing the Wiener index of the so obtained numerical matrix one has novel topological index, referred to briefly as Wiener of Wiener, and labeled as W(W). As shown in Table 3, index W(W) was found to be the best descriptor for regression of Scott's steric contributions of octane isomers.

Moreover in Table 3 we show statistical parameters for Scott's steric contributions of octane isomers for

steric contributions for isomers of octane with selection of topological indices. Indices indicated by asterisk are introduced in ref. 13.

Descriptor F Ref.

W(W) 0.9846 0.331 237.6 *

WW 0.9839 0.339 227.3

WWP 0.9839 0.339 226.8 *

111 0.9813 0.364 195.3

W 0.9806 0.371 187.7

eig W(W) 0.9777 0.398 162.5 *

J 0.9766 0.407 154.8

Shell S2 0.9760 0.412 150.9

eig WW 0.9753 0.419 145.9 *

p) 0.9770 0.417 145.5 *

H 0.9639 0.504 98.3

IIJJ 0.9497 0.593 69.0

R*R 0.9365 0.664 53.5 *

RRW2 0.9008 0.823 32.3

ID 0.8998 0.827 31.9

P2/w2 0.8953 0.844 30.3

X 0.7824 l.l80 11.8

Z 0.7036 1.346 7.4

18 different descriptors, including half a dozen (shown by asterisks) derived from the novel graphical matrix. The message of Table 3 is to point out that while many topological indices may yield simjlar results, nevertheless some are better than others when a particular property is considered. For instance, it is known from the literature that X and Z, which are at the bottom of the table (hence among those shown the

RANDle: CHEMICAL GRAPH THEORY 1213

"worst" descriptors for the property considered) are in fact the best molecular descriptors for the boiling points of alkanes!

As discussed at some length in the manuscript (and outlined in the abstract) novel indices give the best regression for selected physicochemical properties of octanes, hence are superior the already existing indices for the properties considered. Graph Theory has been found a useful tool and has its place in physics, biology, including biochemistry, molecular and cellular biology, proteomics and genomics, ecology, geography, in economics, in psychology, linguistics, social sCiences and In

chemistry.

Acknowledgment The author would like to thank Professor A T

Balaban (Texas A & M at Galveston, TX) for discussions concerning this manuscript and many of his valuable comments.

References I Euler L, Solutio Problematic ad Geometriam Situs

Pertinantis, Academimae Petropolitanae, 8 (1736) 128 .

2 Konig D, Einfiihrung in die Theorie der endlichen lind unendliched Graphes (Chel sea, New York), 1950.

3 Gibbons A, Algorithmic graph theory (Cambridge Univ Press, Cambridge), 1985.

4 Temperley H N V, Graph theory and applications (Ellis Horwood, Chichester), 1981.

5 Randic M & Krilov G, Chern Phys Lett. 272 ( 1997) 115 ; Balaban A T, Rev Roum Chern, 21 (1976) 1049.

6 Read R C, J chern InfCompllt Sci, 23 (1983) 135; Randic M, Int J Quantum Chern Quantum Biol Symp, (1980) 187 ; Randic M, J chem Inf Comput Sci, 35 ( 1995) 357; Randic M Croat chern Acta, 59 (1986) 327; Godson A L, J chern Inf Comput Sci, 20 (1980) 167; Godson A L, Croat chem Acta, 56 (1983) 315; and Lozac ' h N, J chem Inf Comput Sci, 25 ( 1985) 180.

7 Randic M, Acta chim Sloven, 45 (1998) 239; Randic M, Acta Cryst, A34 (1978) 275.

8 Schmalz T G, Klein D J & Sandleback B L, J chem Inf Comput Sci, 32 (1992) 54.

9 King R B, Chern Rev, 101 (2001) 1119; Randic M, Oakland D 0 & Klein D J, J comput Chern, 7 (1986) 35.

10 Matsen F A, Int J Quantum Chem Symp, 8 (1974) 379; Paid us J, J chem Phys, 61 (1974) 5321; Paldus J, Lecture Notes in Chemistry, No. 22 (Springer-Verlag, New York), 1981 ; Shaviu I, Int J quant Chem, 11 (1977) 131.

II Balaban A Tin: Bonchev D & Melenyan 0 (eds), Graph theory approaches to chemical reactivity (Kluwer, Dordrecht). 1994.

12 Corey E J, Rev chem Soc, 25 (1971) 455; Corey E, The logic of chemical synthesis (Wiley, New York), 1989; Zefirov N S & Gordeeva E V, Usp Khim, 56 (1987) 1753.

13 Klein D J, J chem Phys, 75 (1981) 5186; Gordon M & Temple W B in: Balaban AT (ed), Chemical applications of graph theory (Academic Press, London), 1976, pp 299-332.

14 Randic M, Jerman-Blaz ic B, Rouvray D H, Seybold P G & Grossman S C, Int J QlIantum Chem Quantum Bioi Symp, 14 (1987) 245; Randic M in: Rein E (ed), Molecular bases of cancer, Part A: Macromolecular structure, carcinogens, and ollcogens (Liss, New York), 1985, p 309.

15 Randic M & Balaban AT, J chem Inf Comput Sci (in press). 16 Randic M, Novic M & Vracko,M, J Proteome Res, I (2002)

217;Randic M, In! J Quantum Chern, 90 (2002) 848 ;Randic M, Zupan J & Novic M, J chem Inf Comput Sci, 41 (2001) 1339.

17 The manuscript: Randic M & Basak N, Novel graphical matrix and novel distance based molecular descriptors, has been since resubmitted to SAR & QSAR environ Res (in press).

18 Kovacek D, Margetic D & Maksic Z B, J molec Siruct (Theochem) 285 (1993) 195.

19 Randic M, Chem Phys Lett, 38 (1976) 68 ; Randic M, J Am chem Soc, 99 (1977) 444; Randic M, Tetrahedron, 33 (1977) 1905.

20 Schaad L J & Hess B A, Pure appl Chem, 54 (1982) 1097 ; Herndon W C, MA TCH Commull math Comput Chern, 11 (1981) 3.

2 1 Herndon W C, J Am chem Soc, 95 (1973) 2404. 22 Simpson W T , J Am chem Soc, 75 (1953) 597; Simpson W T ,

Theories of electrons in molecules (Prentice-Hall, Eng lewood Cliffs), 1962.

23 Klein D J, Topics curr Chern, 153 ( 1990) 57;Klein D J, J chern Educ, 67 ( 1990) 633; Klein D J & Trinajst ic N, Pure appl Chem, 61 ( 1989) 2107; Klein D J, Pure appl Chern, 5 ( 1983) 299;Klein D J, J chem Edllc, 69 (1992) 691 ;Zivkovic T , Randic M, Klein D J , Zhu H Y & Trinajstic N, J comput Chem, 16 ( 1995) 517; Klein D J, Zhu H, Valenti R & Garcia­Bach M A, Int J quan! Chem, 65 (1997) 421; Zhu H & Klein D J, MATCH Commun math Comput Chem, 31 (1994) 205. Zhu H, Balaban A T, Klein D J & Zivkovic T p, J chem Phys, 101 (1994) 5281; Liu X, Schmalz T G & Klein D J, Chem Phys Lett 188 ( 1992) 550; Seitz W A, Hite G E, Schmalz T G & Klein D J in: King R B & Rouvray D H (eds), Graph theory alld topology in chemistry (Elsevier, Amsterdam), 1987; Klein D J, Hite G E & Seitz W A, J comput Chern, 7 (1986) 443.

24 Klein D J, Schmalz T G, Hite G E & Seitz W A, JAm chem Soc, 98 (1986) 1301. Schmalz T G, Seitz W A, Klein D J & Hite G E, Chem Phys Lett, 130 (1986) 203.

25 Balaban A T & Ivanciuc 0 in: Devillers J & Balaban A T (eds), Topological indices and related descriptors in QSAR and QSPR (Gordon & Breach, Amsterdam), 1999, pp 21-57 ; Basak S C in: Devillers J & Balaban A T (eds), Topological indices alld related descriptors in QSAR and QSPR (Gordon & Breach, Amsterdam), 1999, pp 563-593; Randic M, J chem Inf Comput Sci, 37 (1997) 672; Randic M, J mol Graphics Model, 20 (200 1) 19; Estrada E in: Devillers J & Balaban A T (eds), Topological indices and related descriptors in QSAR and QSPR (Gordon & Breach, Amsterdam), 1999, pp 403-45 3; Estrada E & Molina E in: M V Diudea (ed), QSARlQSPR studies by molecular descriptors (Nova, New York), 2000, pp 91-1 16; Estrada E & Uriarte E, Curr med Chern , 2001 (in print).

26 Randic M in : Schleyer P v R, Allinger N L, Clark T , Gasteiger J, Kollman P A, Schaefer III H F & Schreiner P R (eds), The encyclopedia of computational chemistry (Wiley, Chichester), (1998), pp 3018-3032.

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27 Flower D R, J 11101 Graphics Mode, 16 (1998) 239. 28 Grassy G, Calas B, Yasri A, Lahana R, Woo J, Iyer S,

Kaczorek M, Floc'h R & Buelov R, Nature Biotechnol, 16 (1998)748.

29 Andrade J, Theor Bioi, 197 ( 1999) 193. 30 Katritzky A R. Karelson M & Lobanov Y, CODESSA

(Comprehensive Descriptors for Structural and Statistical Analysis); University of Florida: Gainesville, FL, 1994.

31 For a summary of earlier results obtained with CODESSA see: Katritzky A R, Karelson M & Lobanov Y "Understanding How Chemical Structure Detennines Physical Properties" in which there are summary of 46 publications involving CODESSA (a preprint - courtesy of Professor A. R. Katritzky, Dept. of Chemistry, Univ. of Florida, Gainesville, FL 326 11 ).

32 Agrafiotis D K, J chem In/ Comput Sci, 41 (2001) 159. 33 Liu S S, J chem In/ Comput Sci, 41 (200 I) 321. 34 Galvez J, J chem In/ Comput Sci, 41 (2001) 387. 35 Gao H, J chem In/Comput Sci, 41 (2001) 402. 36 Yaffe D & Cohen Y, J chem InfComput Sci, 4 1 (2001) 463. 37 Estrada E & Molina E, J chem 111/ Comput Sci, 41 (200 I)

791. 38 Burden F R, J chem In/ Comput Sci, 41 (2001) 830. 39 Anderson N L, Esquer-Blasco R, Richardson F, Foxworthy P

& Eacho P, Toxicol Appl Pharmacol, 137 (1996) 75. 40 Randic M, Kleiner A F & DeAlba L M, J chem Ill/ Comput

Sci, 34 (1994) 277. 41 Second Indo-US Workshop on Mathematical Chemistry,

May 3~-June 3, 2000, Duluth, MN; J chem In/ CompL/l Sci 41 (2001) pp 479-701 (total of28 papers).

42 Randic M, Plavsic D & Razinger M, MATCH Commun math Comput Chem, 35 (1997) 243.

Appendix 1

Introductory Graph Theory books: Ore, O. Graphs and Their Uses, Random House: New York:

( 1963). Wilson, J. R. Introduction to Graph Theory, Oliver and Boyd:

Edinburgh (1972). Bondy, J. A.; Murty, U. S. R. Graph Theory with Applications,

MacMillan Press, Ltd: London (1976). Chartrand, G. Graphs as Mathematical Models, Prindle, Weber &

Schmidt, Inc.: Boston 1977. West, D. B. Introduction to Graph Theory. Prentice-Hall, Upper

Saddle River, NJ , 1996. Konig, D. Ein/iihrung in die Theorie der Endlich en !/lui

Unendliched Graphes, Chelsea: New York (1950).

Textbooks and advanced textbooks on GT: Ore, O. Theory 0/ Graphs, Am. Math. Soc. Providence: RI (1962) Harary, F. Graph Theory, Addison-Wesley: Reading, MA (1969) Bucvkley, F.; Harary, F. Distance in Graphs, Addison-Wesley

Pub. Redwood City, CA, 1990. Bursaker, R. G.; Saaty, T. L. Finite Graphs and Networks,

McGraw-Hill, NY, 1965. Harary, F.; Palmer, E. M. Graphical Enumerations. Academic

Press, New York, NY 1973 Tutte, W. T. Graph Theory, Addison-Wesley: Reading, MA 1984

For definitions of graph theoretical terminology see: Essam, J. W.; Fisher, M. E. Rev. Mod. Phys. 1970,42, 271.

Books on Chemical Graph Theory: Chemical ApplicatiollS 0/ Graph Theory (Balaban, A. T. Ed.)

Academic Press, London, (1976) Trinajstic, N. Chemical Graph Theory, (2nd ed.) CRC Press, Boca

Raton, FI 1992. Computational Chemical Graph TheOlY (Rouvray, D. H. Ed.),

Nova Sci. Pub!.: Commack, N. Y. (1990). Merrifield, R. E.: Simmons, H. E. Topological Methods in

Chemistry, John Wiley & Sons, New York, NY, 1989 From Chemical Topology to Three-Dimensional Geometry,

Balaban, A. T. Ed. (Balaban, A. T. Ed.) Plenum, New York, (1997).

Topological Indices and Related Descriptors in QSAR and QSPR (Devilers, 1. ; Balaban, A. T. (Eds.); Gordon & Breach Pub!. Amsterdam (1999)

QSPR I QSAR Studies by Molecular Descriptors, Diudea, M. Y. Ed. Nova Sci .: Pub!. Huntington, NY, 2002.

A Selection of review articles on various topics of CGT: Balaban, A. T. ; Ivanciuc, O. in : Topological Indices and Related

Descriptors in QSAR alld QSPR (Devi lers, J.; Balaban, A. T. (Eds.); Gordon & Breach Pub!. Amsterdam (1999), pp 2 1-57.

Basak, S. C. in : Topological Indices and Related Descriptors in QSAR and QSPR (Devilers, J.; Balaban, A. T., Eds.); Gordon & Breach Pub!.: Amsterdam ( 1999), pp 563-593.

Trinajstic, N.; Klein, D. J.; Randic, M. On Some Solved and Unsolved Problems of Chemical Graph Theory. Int. J. QuantuTll Chem: Quantum Chem. Symp. 1986,20,699-742.

Randic, M.; Trinajstic, N. Notes on Some Less Known Early Contributions to Chemical Graph Theory, Croat. Chern. Acta 1994,67, 1-35.

Balaban, A. T. Chemical Graphs: Looking Back and Glimpsing Ahead. J. Chern. Illf. Compo Sci. 1995, 35, 339-350.

Randic, M. On characterization of Chemical Structure. J. Chern. Inf. Compo Sci. 1997,37,672-687.

Randic, M. The Connectivity Index 25 Years After. J. Mol. Graphics & Modelling 2001, 20,19.

Randic, M. Chemical Structure - What is "She." J. Chern. Educ. 1992, 69,713-718.

Balasubramanian, K. Applications of Combinatorics and Graph Theory to Spectroscopy and Quantum Chemistry. Chern. Rev. 1985,85,599-618.

Pogliani , L. From Molecular Connectivity Indice to Semiempirical Connectivity Terms: Recent Trends in Graph Theoretical Descriptors. Chern. Rev. 2000, 100, 3827-3858.

Balaban, A. T. Solved and Unsolved Problems in Chemical Graph Theory, Annals Discrete Math. 1993,55, 109-126.

Balaban, A. T. Chemical Graphs. Part 49: Open Problems in the Area of Condensed Polycyclic Benzenoids: Topological Stereoisomers of Coronoids and Congeners. Rev. Rourn. Chim. 1988, 33,699-707.

Balaban, A. T. Challenging Problems involving Benzenoid Polycyclics' and Related Systems. Pure Appl. Chern. 1982, 54, 1075-1096.

Balaban, A. T. Is Aromaticity Outmoded? Pure Appl. Chern. 1980,52, 1409-1492.

Appendix 2

We collected numerous questions that cover various topics and results from GT and CGT, which can serve as a test for those who

RANDle: CHEMICAL GRAPH THEORY 1215

think that they know graph theory in order to find out how much they know. We do not supply the answers but they can be found by contacting the literature of CG and CGT. To assist interested readers to find correct answers we listed in the Appendix references that will help to find answers to specific questions.

List of questions to test one's knowledge of Graph Theory and Chemical Graph Theory

There is no significance to the order in which the following questions listed appear. Equally, some questions are more important and some are less important e ither for GT or CGT, some questions are more difficult even if they may be intermingled with easier questions. In reviewing the questi ons we advise that one keep his/her score by adding + I for every correct answer and subtracting a penalty of -2 for every incorrect answer, while having zero fo r unanswered questions. Penalty has been deliberate ly given a greater weight as a wrong answer requires first to eradicate the error and then to learn then correct answer. Observe that several questions have two and three sub-questions, which ought to be considered separately and counted separately. If you answer all questions correctly you can have as much as 100 points and proudly claim to be an expert on GT and CGT. Even a score of 50 would indicate a fair fa mili arity with GT and CGT, but anything below 50 poi nts to lack of familiarity with GT and CGT.

Here is a list of questions: I. What is Clarke's theorem about? Why is this theorem so

important? 2. What is the di ffe rence between the Characteristic polynomial

and Matching polynomial? 3. John Platt made two important contributions to chemical GT.

What are they? 4. Why is Petersen graph of interest in chemistry? What does it

represent in chemistry? 5. When has been published the first book on GT? In what

language? Who was the author? What was his nationality? 6. In what year was published the first book on Chemical GT?

Who was the editor? 7. What is a DID matrix? What structural interpretation has its

leading eigenvalue? 8. What is the elegant algorithm of Gordon and Davidson for? 9. What is a theorem of John and Sachs about? 10. What is the difference between the "conjugated circuits" and

"circuits of conjugation"? Both terms were used in the literature.

II. Consider use of a single variable connectivity index (based on several variables) in structure-property-activity regressions. Does this then represent a simple regression or a multivariate regression analysis?

12. What is the major advantage of orthogonalized molecular descriptors in comparison with standard (non-orthogonal) descriptors?

13. What is the difference between graphical and graph theoretical methodology?

14. Can a graph theoretical index characterize chirality? Gi ve an example if possible or state "not possi ble".

15. What is the difference between a graph theoretical invariant and a topological index?

16. Who introduced Distance matrix in GT? 17. What are the Kronecker and the Hadamard product of

matrices?

18. What is Dellaney triangulation of a map? How is it constructed? Why may it be of interest in chemistry?

19. What is the "monster" graph? How big is the monster graph (how many vertices)?

20. How was the Wiener index defined by Harry Wiener and how is it related to the distance matrix?

2 1. What is a structural interpretation of the Wiener index? 22. What is the difference between )(jer & Hall' s valence

con nectivity index and the relatively recently proposed variable connectivity index?

23. Why is phenanthrene more aromatic than anthracene? 24. Is it true that n-alkane always have the highest BP (Boiling

point) among 'all isomers having the same n? If yes - why yes? [f no - why no?

25. What is the difference between a graph matrix (such as the adjacency matrix, the distance matrix) and graphical matrices of graphs?

26. It is well known that one can easi ly obtain the adjacency matri x fro m a di stance matrix. Can one construct (not looking at a picture of a graph) the distance matrix from the adjacency matri x?

27. Do topological indices have to have phys ical meaning? If the answer is yes, then why?

28. [s partial ordering of interest in chemistry? Give an illustrati on if the answer is positi ve.

29. What is the difference between molecular topological indices and molecular topographic indices?

30. What is the major result of a work of Collatz and Sinogowitz?

3 1. What is "graph Reconstruction" problem? Who and when proposed the problem? Has the problem been solved?

32. What is the contribution of Joshua Lederberg (Nobel Laureate) to chemical notation?

33. What is the contribution of Vlado Prelog (Nobel Laureate) to chemical notation?

34. What are Ugi 's BE matrices? 35. What are the uses of canonical labeling of vertices of graphs:

To solve Graph Isomorphi sm problem? To solve Graph automorphism problem? Both?

36. Who was first to publish a paper illustrating subgraphs contributing to the construction of the characteristic polynomial?

37. Who and when was the first to use term "graph" for mathematical objects known today as graphs?

38. Who was first to consider enumeration of graphs? 39. Who and when published the first paper on enumeration of

chemical isomers? In what journal? In what language? 40. What are (vertex and edge) transitive graphs? Why are they

of interest in chemistry? Who and when introduced first such graph in chemistry?

41. What is the size of the smallest edge (but not vertex) transitive graph? Who constructed this graph?

42. What graphs are cages labeled as cages? Is the Coxeter graph (vertex and edge transitive cubic graph on n = 28 vertices) a cage?

43. What is graph center? What is graph diameter? What is girth of a graph?

44. Who classified graph algorithms as NP (non-polynomial ) and P (polynomial)?

45 . Is enumeration of paths of different length in a graph solved by a non-polynomial (NP) or polyno;nial algorithm?

1216 INDIAN J CHEM, SEC A, JUNE 2003

46. Is enumeration of walks of different length in a graph solved by a non-polynomial (NP) or polynomial algorithm?

47. What is the detour matrix? Who introduced the detour matrix in GT? Does it have another name?

48. Can two non-isomorphic graphs have identical distance matrix? Can two non-isomorphic graphs have identical detour matrix?

49. How is Hosoya's topological index constructed? 50. What is the Frobenius theorem that defined bounds on the

leading eigenvalue of a matrix? 51. What are "migrating" sextets? 52. Has Klaus Ruedenberg published a paper that can be viewed

as CGT paper? 53. Has Hans Primas published a paper that can be viewed as

CGT paper? 54. Has C. A. Coulson published a paper that can be viewed as

CGT paper? Has Linus Pauling (doubly Nobel Laureate) published a paper that

can be viewed as CGT paper?

Has G. Wheland published a paper that can be viewed as CGT paper?

Has Rudolf Marcus (Nobel Laureate) published a paper that can be viewed as CGT paper?

55. What is the crossing number? What is the crossing number of 4-D cube?

56. Are fullerene graphs planar graphs? 57. What is Laplace matrix and what for it is used in GT and

CGT? 58. Are the row sums of a symmetric matrix invaraints? Is the sum of all matrix elements of a sy mmetric matrix an

invariant?

59. Should a shape index depend on molecular size (number n of vertices)? Do Kier's K2 and K3 depend on n?

60. Do we have enough topological indices? If yes - why yes? If no - why no?

We list here sources for checking the answers to the 60 questions li sted in the text rather than providing the answer in order that an effort is made to find the answer. For instance, on question 5 we could have li sted easily the name of the author, the title of the book (which would reveal the language), indicate the nationality of the author, and even giving some biographical details (for instance, the author of the book has committed suicide during WW II). But that would be counterproductive to our desire that readers do the investigative research. Hence, we give hints where the answers could be found and at a later time we plan to give the answers and accompanying information .

Answers to question can be found in: Q I: Journal a/Graph Theory Q 2: See: Trinajstic, N. Computing the Characteristic

Polynomial of a Conjugated System Using the Sachs Theorem. Croat. Chem. Acta 1977,49, 593-633.

Q 3: See: Platt, J. R. in Handbuch der Physik (Fliigge, S. Ed.), Springer-Verlag: Berlin (1961), pp. 205-209; and: Platt, J. R. Prediction of isomeric differences in paraffin properties. J. Phys. Chem. 1952,56,328-336.

Q 4: See: Dunitz, J. D. ; Prelog, V. Allgew. Chem. 1968, 80, 700; Balaban, A. T.; Farca~iu , D.; Banicii, R. Rev. Roum. Chim. 1966, II, 1205.

Q 5: See: Graph Theory 1736 - 1936.

Q 6: See the first edition of Konig, D. Einfiihrung in die Theorie der Elldlichell und Unelldliched Graphes, which was reprinted by Chelsea: New York in 1950.

Q 7: See: Randic, M.; Kleiner, A. F.; DeAlba, L. M. J. Chem. In! Comput. Sci. 1994, 34, 277.

Q 8: See: Gordon, M. ; Davison, W. H. T. 1. Chem. Phys. 1952, 20,428.

See also: Randic, M. Aromaticity of Polycyclic Conjugated Hydrocarbons. Chem. Rev. (in print).

Q 9: See: John, P.; Sachs, H. Calculating the Numbers of Perfect Matchings and of Spanning Trees, Pauling' s Orders, the Characteristic Polynomial, and the Eigenvectors of a Benzenoid system. Topics Curro Chem. 1990, /53, 146-179.

Q 10: See: Randic, M. Aromaticity of Polycyclic Conjugated Hydrocarbons. Chem. Rev. (in print).

Q II: Some think "yes" (Zefirov, N. S.; Palyulin, V. A. QSAR for boi ling points of "small" sul tides. Are the "High­Quality Structure-Property-Activity Regressions" the real high quality QSAR model. J. Chem. In! Compul. Sci. 2001 ,4 /, 1022-1U27) but some think the correct answer is "no" (see: Randic, M. Variable Molecular Descriptors and High-Quality Regression Analysis. Understanding and Misunderstandings. J. Chem. In! Comput. Sci. (submitted).

Q 12: See: Randic, M. Resolution of ambiguities in structure­property studies by use of orthogonal descriptors. 1. Chem. In! Comput. Sci. 1991, 31 , 3 11 -320; Randic, M. Orthogonal molecular descriptors. New.l. Chem. 1991 , 15, 517-525; Randic, M. Fitting of non-linear regressions by orthogonalized power series. J. Compul. Chem. 1993, 14, 363-370.

Q 13: For illustrat ions of a graphical approaches see: Randic, M. Graphical Enumeration of Conformations of Chains. Int. J. Quantum Chem: Quantum Bioi. Symp. 1980, 7, 187-197. Klavzar,S.; Zigert, P.; Gutman, I., Theochem (submitted). Randic, M. Polycyclic Aromatic Hydrocarbons (submitted).

Q 14: See: Liang, c.; Mislow, K. J. Math. Chem. 1995, 18, 1-24; Randic, M. ; Razinger, M. J. Chem. In! Comput. Sci. 1996, 36, 429-442; Randic, M. 1. Chem. Ill! Comput. Sci. 2001 ,41,639-649. Golbraikh, A.; Bonchev, D.; Tropsha, A. J. Chem. In! Comput. Sci. 2001 , 41,147- 158. Schultz, H. P.; Schultz, E. B.; Schultz, T. P. 1. Chem. In! Comput. Sci. 1995, 35, 864-870.

Q 15: See any of many review articles on topological indices, which are more popular (but somewhat incorrect) name for graph theoretical invariants also referred to as molecular descriptors.

Q 16: See: Harary, F. Graph Theory, Addison-Wesley: Reading, MA (1969)

Q 17: See any textbook on Linear Algebra or MATLAB Manual. Q 18: Dellaney triangulation is dual of a map which is divided in

regions such that all points with a region are closer to the center of the region than to any other neighboring center. See any textbook on advanced geometry.

Q 19: See: Randic, M.; Oakland, D. O.; Klein, D. 1. J. Compul. Chem. 1986, 7, 35. Zivkovic, T. Croat. Chem. Acta

Q 20: See: Wiener, H. Prediction of isomeric differences In

paraffin properties. J. Am. Chem. Soc. 1947,69, 17-20.

RANDle: CHEMICAL GRAPH THEORY 121 7

See: Hosoya, H. Topological index. A new ly proposed quantity characterizing the topo log ical nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. l apan, 1971 , 44,2332-2339.

Q 21 See: Randic, M.; Zupan. 1. On Structura l Interpretat ion of Topological Indices, In : Proc. Wiener Melllorial Conference. Athens 200 1 (R. B King and D. H. Rouvray, Eds.); Rand ic, M.; Zupan, J . On interpretat ion of well ­known topological indices. 1. Chen!. In! Comput. Sci. 2001 ,41,550-560.

Q 22:See: The Connectivity Index 25 Years after (J. Molecular Graphics alld Modelling, 2001 ,20, 19-35).

Q 23: See: Randic, M. Aromaticity of Polycyclic Conjugated Hydrocarbons, Chelll. Rev. (in press). Also: Randic, M. l . Am. Chefl/. Soc. 1976,

Q24: See Randic M; Wilkins. C. L. On graph theoretical basis for ordering of structures. Chelll . Phys. U fl . 1979, 63, 332-336: Randic M; Wilkins, C. L. Graph theoret ica l ordering of structu res as a basis for systematic searches for regularity in molecular data. l. Phys. Chem. 1979, 83, 1525- 1540.

Q 25: See: Randic, M .; Plavs ic, D.; Razinger, M. Double In variants. MATCH-Comlllun. Math. COllljJLIl. Chem. 1997,35,243-259.

Q 26: See: Kunz, M. An equi valence relation between distance and coordinate matrices. MATCH - Comlllull. Math. Comput. Chen!. 1995,32, 193-204.

Q 27: See: Randic, M.; Zupan , 1. On the structural interpretation of topological indices, in : Topology ill Chemistry- Discrete Mathematics of Molecules (Rouvray, D. H.; King, R. B., Eds.), Horwood Pub!.: Ch ichester, England, 2002, pp. 249-29 1.

Estrada, E. The Physicochemical Interpretatio n of Molecul ar Connectivity Indices, l . Phys. Chell!. (in press)

Q 28: See: Klein, D. 1. Prolegomenoll on Partial Orderings in Chemistry, MATCH - Commun. Math. Compllt. Chem. 2000,42, 7; Randic, M. Vracko, M.; Novic, M. Basak, S. C. MATCH - Com/1/un. Math. Comput. Chern. 2000, 42. 18 1.

Q 29: See: Randic, M.; Razinger, M. Molecular Topographical Indices. 1. Chem. In! Comput. Sci. 1995, 35, 140- 147; Also: Estrada E. Characterization of 3D Molecular Structure. Chem. Phys. Lefl . 2000, 319, 7 13-718.

Q 30: See: Cvetkovic, D. M. ; Doob, M. ; Sachs, H. Spectra of Graphs, 3rd ed. Huthig GmbH - 1. A. Barth Verlag, Heidelberg, 1995;

Collatz, L. ; Sinogowitz, U. Abh. Math. Sem. Ulliv. Hamburg 1957, 21, 63.

Q 3 1: See: O'Neil, P. V. Ulam's Conjecture and Graph Reconstruction. Amer. Math. Montly, 1970, 77, 35-43.

Q 32: See: Lederberg, 1.; Sutherland, G. 1.;Buchanana, B. G.; Feigenbaum, E. E.; Robertson, A. V.; Duffield , A. M.; Djerassi, C. Application of Artificial Intellige nce for Chemical Inferences. I. Ther Number of Possible Organic Compounds. Acyclic Structures Containing C, H, 0, and N. l . Am. Chern. Soc. 1969,91 ,2973-2976 ..

Q 33: See: Nobel Laureates in Chemistry 1901 - 1992 (Laylin K lames, Ed.) Amer. Chem. Soc. Washington, D. c., 1993; Cahn, R. S. ; Ingold, C. K; Prelog, V. Angew. Chem. Int. Engl. Ed. 1966, 5, 385.

Q 34: See: Dugundji , J.; Ugi, I. Topics Curro Chem.1973, 39, 19.

Q 35: See: Read, R. c.; Corneil, D. G. The Graph Isomorphi sm Di sease, l . Graph Theory 1977, 1,339.

Q 36: See: Coulson , C. A. Proc. Cambridge Phil. Soc. 1950, 46, 202.

Q 37: See: Rouvray, D. H., l . Mol. Stmct. 1989, 185, I ;Bell , E. T. Men of Mathelllatics; See also an article entitled Chemistry and Algebra by Sy lvester, 1. J . Nature, 1877-1878, XVll. pp. 284, 309.

Q 38: See a chapter on graph enumeration in: Biggs, R; Lloyd, E. K.; Wilson, R. J. Graph Theory 1736 - 1936, Clarendon Press: Oxford, 1976.

Q 39: See: Balaban , A. T. Enumi!fatioll of Isomers. In Chemical graph theory. Abacus Press, Gordon & Breach: New York, 1990, p. 177-234.

Randic M.; Trinajstic, N. On some Less Known Graph Theory Contributions, Croat. Chell!. Acta 1994,67, 1-35.

Q 40: See: Trinajstic, N. Chemical graph theory, (2nd ed .) CRC Press: Boca Raton, FI 1992.

Q 41: See: Pi sanski, T.; Randic, M. Bridges between Geometry and Graph Theory; in : Geometry at Works (Papers in Applied Geometry), Gorini C. A., Editor, MAA (Mathematical Association of America) Notes Number 53, Washington, D. c.: 2000 or:

Bondy, J. A.; Murty, U. S. R. Graph Theory with Applications, MacMillan Press, Ltd: London (1976).

Q 42 : See: Balaban, A. T. 1. Combinatorial Theory, Ser. B 1972, 12, I ; Balaban, A. T. Rev. Roum. Math. Pures Appl. 1973, 18, 1033; Also: Wong, P. K Cages - a survey. l . Graph Theory. 1982, 6, 1-22.

Q 43: See: Harary. F. Graph Theory, Addison-Wesley: Reading, MA ( 1969);Bonchev, D.; Balaban, A. T.; Randic, M. Int. l . QuallIum Chem. 1981, 19,61.

Q 44: See: Miller, R. E.; Thatcher, J. W. Complexity of Computer Computations. Plenum Press, 1972, pp. 85-103.

Q 45: See: Miller, R. E.; Thatcher, J. W . Complexity of Computer Computations. Plenum Press, 1972, pp. 85- 103.

Q 46: Harary, F. Graph Theory, Addison-Wesley: Reading, MA ( 1969).

Q 47 : See: Trinajstic, N.; Nikolic, S.; Lucic, B.; Amic, D.; Mihalic, Z. The Detoru Matri x in Chemistry. 1. C/leln. In! Comput. Sci. 1997, 37, 631-630;Nikolic, S.; Trinajstic, N.; luric, A.; Mihalic, Z. The Detoru Matrix and the Detour Index of Wei ghted Graphs. Croat. Chem. Acta 1996, 69, 1577-1591 ;Lukovits, I. The Detour Index. Croat. Chem. Acta 1996, 69, 873-882; Lukovi ts, I.; Razinger, M. On the Calculation of the Detour Index, l . Chem. In! Comput. Sci. 1997, 37, 283-286.; Amic, D. ; Trinajstic, N. On the Detour Matrix. Croat. Chem. Acta 1995,68,53-62.

Q 48: See: Randic, M.; DeAlba, L. M. ; Harri s, F. E. Graphs with the same Detour Matrix . Croat. Chem. Acta 1998, 71 , 53.

Q 49: See: Hosoya, H. Topo logical index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. lpn. 1971 , 44,2332-2239 .

Q 50: See a textbook on Linear Algebra Q 51: Clar , E. The Aromatic Sextet J. Wiley & Sons: London

1972 Q 52: See: Ruedenberg, K l . Chem. Phys. 1954, 22, 1878;

Ham, N. S.; Ruedenberg, K l . Chem. Phys. 1958, 29, 1215.

1218 INDIAN J CHEM, SEC A, JUNE 2003

Q 53: See: Glinthard. H. H.; Primas, G. Helv. Chill/. Acta 1956, 39, 1645.

Q 54: . See: Coulson, C. A. Proc. Call/bridge Phil. Soc. 1950,46, 202: Pauling, L. 1. Chem. Phys. 1933, I, 280; Wheland, G. W. 1. Chem. Phys. 1933, 3, 356: Marcus, R. A. 1. Chem. Phys. 1965, 43,2643.

Q 55: See: Harary, F. Graph Theory, Addi son-Wesley : Reading, MA ( 1969).

Q 56: All polyhedral graphs are planar Q 57 : See, for instance, Mohar, B. Laplace eigenvalues of graphs

- a survey. Discrete Math. 1992, 109, 171-183; Mohar, B. Some applications of Laplace eigenvalues of graphs.

Hahn. G.: Sabidussi. G. (Eds.) Graph Symmetry: Algebraic Methods alld Applications, (NATO ASI series, Series C, Mathematical and Physical sciences, vol. 497 ). Dordrecht; Boston; London: Kluwer, 1997, pp. 225-270.

Q58: True only for regular graphs in which case the row sums are constant a,nd represent trivial invari ant (of little intereest).

Q 59: See: Randic, M. Novel Shape Descriptors for Molecular Graphs. 1. Chem. Inf. Call/put. Sci. 2001, 41, 607-613.

Q 60: The answer is "yes" if you want to freeze the development of chemical graph theory as a scientific discipline; if you think that science is not static, the answer is an emphatic "no."