Indian Journal of 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 , 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 , 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 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 so short extract of this paper: "The data set used in this numerically characterize a structure - hence a need study is based on the reductive amination reaction ... for topological indices and other structural invariants. and is utilized for the construction of structurally diverse druglike molecules with useful Recent accomplishments in use of topological pharmacological properties . . . For demonstration indices as molecular descriptors purposes, 300 primary and secondary amines and 300 We collected a dozen illustrations from the aldehydes were selected at random . . . and were used literature in which topological indices have been used to generate a virtual library of 90,000 products . . . 7 39 for diverse problems of chemistrl . , to examine the Each compound in the 90,000-membered library was papers and then judge if topological indices are characterized by an established set of 117 topological useful. descriptors, which were subsequently normalized and Flower27 in his paper on DISSIM, a computer decorrelated using principal component analysis, program which addresses the problem of selecting resulting in an orthogonal set of 23 latent variables diverse subsets from larger collections of chemical which accounted for 99% of the total variance of data compounds considered relationships of 159 topological indices (listed in his Table 1) of which 39 Liu and collaborators33 reported on a molecular were used in his CACS (Computer-Aided Compound electronegativity distance vector based on 13 atomic Selection) protocol. types, called MEDV -13, as a descriptor for predicting Lahana and coworkers28 by selecting four from two the biological activities of molecules based on QSAR. dozen topological indices for screening a Their conclusion was that the study on steroids shows combinatorial library of some 280,000 virtual that the performance of the MEDV-13 method has decapeptides were able, by selecting "windows" of comparability with the previous methods containing allowed values for individual topological indices and 3D QSAR, and the peptide study gives a high quality nine other molecular descriptors, to reduce this of the QSAR model based on the MEDV -13 method. enormous library to 26 compounds on which they However, the MEDV -13 method only employs performed more advanced calculations. Finally they information about an element type, valence selected five compounds and synthesized them electron state, and type from the 20 finding that one compound had an molecular topology and requires no information immunosuppressive activity that was almost hundred related to 3D structures or physicochemical properties times more active than the lead compound! or molecular alignment. So, the MEDV -13 descriptor Andrade and coworkers29 used novel topological is fast, easy to use, reproducible and predictable one indices to characterize t-RNA. They proposed for the QSAR studies. "weighted structural descriptors-closely related to Galvez and collaborators34 tried to combine Randic connectivity and Balaban distance indices-as parameters of semiempirical (quantum chemical) distinctive characteristics of each structure. Molecules calculations with topological indices in order to arri ve were characterized by a set of weighted structural at any discriminant function for antibacterial activity. descriptors and classified by a clustering method and They found that two quantum chemical descriptors discriminant function analysis. Two main groups of (QDs) played important role for discriminant function tRNAs that correspond to the biosynthetic amino acid 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 connectivity indices (I XV, 3 xv. 4 XV), molecular weight, (TIs). The utility of expressing QDs "as a function of and temperature as an input parameters. . . .The the TIs is to the extent that their predictive capability performance of the QSPR for temperature-dependent is for larger sets of compounds for which the vapor pressure, which was developed from a simple calculation of TIs is much faster and easier than the set of molecular descriptors, displayed accuracy of calculation of QOs .. .". They concluded: "Using the better than or well within the range of other avrulable multilinear regression and the LDA (linear estimation methods. discriminant analysis), a pattern of topological Estrada and Molina37 considered the classification simjlarity of antibacterial activity has been obtained. of antibacterial activity of 2-furylethylene derivatives. This pattern has been applied successfully for the According to them: Topographic (3D) molecular search of drugs that together with other connectivity indices based on molecular graphs pharmacological activities can also show antibacterial weighted with quantum chemical parameters are used activity. An added advantage is that this screening can in QSPR and QSAR studies. These descriptors were be carried out using large databases and with low compared to 2D connectivity indices (vertex and edge time-consuming." ones) and to quantum chemical descriptors in Ga035 reported that the genetic algorithm was very modeling partItIon coefficient (log P) and efficient method for variable selection or antibacterial activity of 2-furylethylene derivatives. In optimization. Besides available topological indices he describing log P the 3D connectivity indices produced also used BCUT metric (an extension of Burden's a significant improvement (more than 29%) in the parameters, which are based on a combination of the predictive capacity of the model compared to those atomic number for each atom and a description of the derived with topological and quantum cherrucal nominal bond type for adjacent and non-adjacent descriptors. The best linear discrimjnant model for atoms and incorporates both connectivity information classifying antibacterial activity of these compounds and atomic properties). He made binary QSAR was also obtained with the use of 3D connectivity analysis of Carbonic Anhydrase II Inhibitors. "The indices. The global percent of good classification best binary QSAR model was obtained with a obtained with 3D and 20 connectivity indices as well combination of 23 molecular descriptors including as quantum chemical descriptors were 94.1,91.2 and . . . d (2 0 v 1 v d 2 V) f our connectIvIty 111 exes X, X, X, an X, 88.2 respectively. three shape indexes CZK, 1Ka, 3Ka), log P(o/w), and 15 We included this paper as it clearly shows that BCUTs .. . The cross-validated accuracy is 91 % on graph theoretical approaches are not lirruted to 20 active compounds, 92% on inactive compounds, and objects (graphs) but can be extended to 3D objects 91 % for all compounds. Thus the predictive power of (embedded graphs). To quote from the introduction of the binary QSAR model is quite high." He also made the paper of Estrada & Molina: An important step binary QSAR analysis of Estrogen Receptor Ligands forward in the development of graph-based molecular and obtained sirrular results: 'The best binary QSAR descriptors has been the definition of topographic model was obtained with a combination of 24 descriptors. These kinds of molecular descriptors are molecular descriptors including four connectivity based on molecular graphs with appropriate weights indexes (0 X, 1 X, 2 X, and 1 XV), two shape indexes (IKa, to account for 3D molecular features. The pioneering 2Ka), flexibility index <1>, log P(o/w), and 16 BCUTs works in this direction were done by Milan Randic at ... The cross-validated accuracy is 73% on active the end of 1980s. In these works Randic proposed the compounds, 90% on inactive compounds, and 88% use of topographic distance matrices, first based on for all compounds. " graph embedded on a hexagonal lattice and then on a Yaffe and Cohen36 reported on estimation of vapor 3D diamond lattice. pressure using topological indices as descriptors. Here Burden38 considered a large Benzodiazepine Data is an extract from their Abstract: A neural network Set (245 compounds), Muscarine Data Set (162 based quantitative structure-property relationship compounds) and Toxicity Data Set (277substituted (QSPR) was developed for the vapor pressure­ benzenes). Here is an extract from his paper: Gaussian temperature behavior of hydrocarbons based on a data processes method (GPM) constitutes a method of set for 274 compounds. The optimal QSPR model was solving regression problems. The usual coefficients or developed based on 7-29-1 back-propagation neural weights associated with other regression methods are network architecture using valence molecular absent and an exact Bayesian analysis is RANDle: CHEMICAL GRAPH THEORY 1211 accomplished using matrix manipulations. . . A classification biological actIvItIes from models of combination of five set of easily computed indices structural similarity; comparison of a neural net-based were employed in this work: the well studied Randic QSAR algorithm with holograms and multiple linear index (R); the valence modification to the Randic regression - based QSAR; graph theoretical analysis index by Kier and Hall (K); and an atomistic (A) of tunneling electron transfer in large polycyclic index developed by Burden which has now been aromatic hydrocarbon networks; on distance related enhanced by recognition of aromatic atoms and indexes; an extension of the Wiener index and the hydrogen atom donors and acceptors (B) . . . Two "overall Wiener index;" interpretation of well-known further indices have been added the first counts the topological indices, and structural interpretation of number of rings of various sizes (G) and the second several distance-related topological indices; counts some common functional groups (F) .. . The characterization of DNA primary sequences based on four type of index, R, K, A, and B, are average distances between bases, and characterization complementary, and we have shown in previous of DNA by triplet of nucleic acid bases; illustration of studies that their combination yields better QSAR CODESSA-based theoretical QSPR model with models than the individual indices alone. variable· molecular descriptors based on distance Now if all this is not sufficient to convince readers related matrices; novel shape descriptors for that topological indices are useful in chemistry and molecular graphs and use of graph shells as molecular found applications in quite diverse problems then I descriptors; characterization of 2D chirality; and would like to draw attention of readers to a paper of several papers on the use of the variable connectivity mine written in collaboration with Novic and index and the hierarchical approach to QSAR. l6 Vracko • In this paper the authors explore the characterization of 2-D electrophoresis proteomics Real world chemistry-what is it? maps by certain structural invariants derived from In the past some cntIcs of CGT were matrices constructed by considering for all pair of "complaining" that researchers in GT are mostly spots in a proteomics maps the shortest (Euclidean) preoccupied with hydrocarbons. Hydrocarbons are distances and distances me red along zigzag lines also part of the "real-world chemistry". If not why connecting protein spots of the neighboring not? In Table 2 are listed a few hydrocarbons and abundance. This paper is a sequel to previous papers their properties just to remind readers that in which we outlined the idea of characterizing 2-D hydrocarbon chemistry, including fullerenes that may proteomics maps by graph-theoretical descriptors. To or may not be decorated with hydrogens, may be as illustrate the approach we selected data of Anderson 39 fascinating as any branch of chemistry. Fullerenes are et al. on protein abundance in mouse liver under even a better illustration of simple chemical graphs in series of dose of peroxisome proliferator L Y 1711883. view that usually hydrogens are not present and need We found strong linear correlation between the not be suppressed! experimentally applied doses and the leading 40 What critics overlook in objecting to modeling eigenvalue of Distance/Distance type matrix CGT on hydrocarbons is to see that this is only the constructed for the experimental proteomics maps. first stage in developing novel molecular descriptors and models. If a new topological index is not useful in Discussion characterizing selected properties of hydrocarbons, Readers will have to judge whether the utility of they are probably even less suitable for characterizing topological indices has been demonstrated "beyond of properties of heteroatomic compounds. Once the reasonable doubt." Observe that half a dozen descriptors are found useful they, as a rule, are "cases" have been taken from not only a single generalized to extend to molecules having journal, but a single volume of that journal. Moreover, heteroatoms. The list of the dozen papers outlined in in the same volume of J chem In! Comput Sci. that we the previous section has numerous such illustrations. screened 28 papers41 are published from the Second Indo-US Workshop on , held Concluding remarks in Duluth, MN, in May 2000 which is "crowded" with We will end with a brief outline of the manuscript papers on application of GT to Chemistry, including that was the reason behind this review. The manuscript numerous publication on novel topological indices. considers several novel topological indices for octane For example, there are papers on: criteria for isomers (but equally applicable to other molecules), 1212 INDIAN J CHEM, SEC A, JUNE 2003

Table 2-Diverse properties of different hydrocarbons Formula Name Property CJ2HIO heat transfer agent; fungistat for oranges

C 1sHI8 Guaiazul ene anti-inflammatory; anti-ulcerative

C I9H32 Tridecylbenzene detergent; forms stable foam in the presence of fat

C I9H40 Pristane lubricant, anti-corrosion agent

C22 H I4 Pentacene large crystal semiconductor

C 21 HI6 3-Methylcholanthrene experimental use in cancer research C2) H46 Muscalure sex pheromone

C30HSO Squalene oil, agreeable odor; bactericide

C30H62 Squalane lubricant; transformer oil; perfume fixative; skin lubricant C.jQHsl Lycopene carotenoid occurring in ripe fruit C4o Hs6 a-Carotene vitamin A precursor C4o H68 Phytofluene polyene hydrocarbon widespread in the vegetable kingdom Ci soHI86 Hexabenzocoronene derivative liquid crystal (component for photovoltaic films) all derived from a new graphical matrix. Matrix Table 3-The regression coefficient for quadratic regressions of elements of a graphical matrix, in contrast to the steric contributions for isomers of octane with selection of topological indices. Indices indicated by asterisk are introduced in traditional graph theoretical matrices (such as the ref. 13. adjacency matrix, distance, matrix, detour matrix, Wiener and Hosoya matrices, etc.) are not numerical, Descriptor F Ref. but are defined through qualified subgraphs of a W(W) 0.9846 0.331 237.6 * graph considered. Graphical matrices have been for WW 0.9839 0.339 227.3 the first time introduced only a few years ago, and besides the seminal paper42, the paper we are talking WWP 0.9839 0.339 226.8 * about is the only other manuscript considered this 111 0.9813 0.364 195.3 17 W 0.9806 0.371 187.7 subject • Graphical matrices are not widely known even among chemical graph theory, being published eig W(W) 0.9777 0.398 162.5 * in less available and less read mathematical journals. J 0.9766 0.407 154.8 So this particular paper was the first opportunity for a Shell S2 0.9760 0.412 150.9 wider circle of readers interested in structural eig WW 0.9753 0.419 145.9 * chemistry to hear about graphical matrices! p) 0.9770 0.417 145.5 * Because matrix elements of graphical matrix are H 0.9639 0.504 98.3 non-numerical the first step in applications is to arrive IIJJ 0.9497 0.593 69.0 at a numerical representation for such matrices, which R*R 0.9365 0.664 53.5 * is based on a selection of a particular graph invariant RRW2 0.9008 0.823 32.3 to serve as "transition" to a numerical matrix. Once a ID 0.8998 0.827 31.9 numerical matrix is constructed, one has choice to P2/w2 0.8953 0.844 30.3 select any of already known procedures to extract X 0.7824 l.l80 11.8 structural invariants from such a matrix. Thus if one Z 0.7036 1.346 7.4 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 18 different descriptors, including half a dozen obtained numerical matrix one has novel topological (shown by asterisks) derived from the novel graphical index, referred to briefly as Wiener of Wiener, and matrix. The message of Table 3 is to point out that labeled as W(W). As shown in Table 3, index W(W) while many topological indices may yield simjlar was found to be the best descriptor for regression of results, nevertheless some are better than others when Scott's steric contributions of octane isomers. a particular property is considered. For instance, it is Moreover in Table 3 we show statistical parameters known from the literature that X and Z, which are at for Scott's steric contributions of octane isomers for the bottom of the table (hence among those shown the RANDle: CHEMICAL GRAPH THEORY 1213

"worst" descriptors for the property considered) are in 14 Randic M, Jerman-Blazic B, Rouvray D H, Seybold P G & fact the best molecular descriptors for the boiling Grossman S C, Int J QlIantum Chem Quantum Bioi Symp, 14 (1987) 245; Randic M in: Rein E (ed), Molecular bases of points of alkanes! cancer, Part A: Macromolecular structure, carcinogens, and As discussed at some length in the manuscript (and ollcogens (Liss, New York), 1985, p 309. outlined in the abstract) novel indices give the best 15 Randic M & Balaban AT, J chem Inf Comput Sci (in press). regression for selected physicochemical properties of 16 Randic M, Novic M & Vracko,M, J Proteome Res, I (2002) octanes, hence are superior the already existing 217;Randic M, In! J Quantum Chern , 90 (2002) 848;Randic M, Zupan J & Novic M, J chem Inf Comput Sci, 41 (2001) indices for the properties considered. 1339. Graph Theory has been found a useful tool and has its 17 The manuscript: Randic M & Basak N, Novel graphical place in physics, biology, including biochemistry, matrix and novel distance based molecular descriptors, has molecular and cellular biology, proteomics and been since resubmitted to SAR & QSAR environ Res (in genomics, ecology, geography, in economics, in press). 18 Kovacek D, Margetic D & Maksic Z B, J molec Siruct psychology, linguistics, social sCiences and In (Theochem) 285 (1993) 195. chemistry. 19 Randic M, Chem Phys Lett, 38 (1976) 68; Randic M, J Am chem Soc, 99 (1977) 444; Randic M, Tetrahedron, 33 (1977) Acknowledgment 1905. 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27 Flower D R, J 11101 Graphics Mode, 16 (1998) 239. Books on Chemical Graph Theory: 28 Grassy G, Calas B, Yasri A, Lahana R, Woo J, Iyer S, Chemical ApplicatiollS 0/ Graph Theory (Balaban, A. T. Ed.) Kaczorek M, Floc'h R & Buelov R, Nature Biotechnol, 16 Academic Press, London, (1976) (1998)748. Trinajstic, N. Chemical Graph Theory, (2nd ed.) CRC Press, Boca 29 Andrade J, Th eor Bioi, 197 (1999) 193. Raton, FI 1992. 30 Katritzky A R. Karelson M & Lobanov Y, CODESSA Computational Chemical Graph TheOlY (Rouvray, D. H. Ed.), (Comprehensive Descriptors for Structural and Statistical Nova Sci. Pub!.: Commack, N. Y. (1990). Analysis); University of Florida: Gain esville, FL, 1994. Merrifield, R. E.: Simmons, H. E. Topological Methods in 31 For a summary of earlier results obtained with CODESSA Chemistry, John Wiley & Sons, New York, NY, 1989 see: Katritzky A R, Karelson M & Lobanov Y From Chemical Topology to Three-Dimensional Geometry, "Understanding How Chemical Structure Detennines Balaban, A. T. Ed. (Balaban, A. T. Ed.) Plenum, New Physical Properties" in which there are summary of 46 York, (1997). publications involving CODESSA (a preprint - courtesy of Topological Indices and Related Descriptors in QSAR and QSPR Professor A. R. Katritzky, Dept. of Chemistry, Univ. of (Devilers, 1. ; Balaban, A. T. (Eds.); Gordon & Breach Pub!. Florida, Gainesville, FL 326 11 ). Amsterdam (1999) 32 Agrafiotis D K, J chem In/ Comput Sci, 41 (2001) 159. QSPR I QSAR Studies by Molecular Descriptors, Diudea, M. Y. 33 Liu S S, J chem In/ Comput Sci, 41 (200 I) 321. Ed. Nova Sci.: Pub!. Huntington, NY, 2002. 34 Galvez J, J chem In/ Comput Sci, 41 (2001) 387. 35 Gao H, J chem In/Comput Sci, 41 (2001) 402. A Selection of review articles on various topics of CGT: 36 Yaffe D & Cohen Y, J chem InfComput Sci, 4 1 (2001) 463. Balaban, A. T. ; Ivanciuc, O. in : Topological Indices and Related 37 Estrada E & Molina E, J chem 111/ Comput Sci, 41 (200 I) Descriptors in QSAR alld QSPR (Devilers, J.; Balaban, A. 791. T. (Eds.); Gordon & Breach Pub!. Amsterdam (1999), pp 38 Burden F R, J chem In/ Comput Sci, 41 (2001) 830. 2 1-57. 39 Anderson N L, Esquer-Blasco R, Richardson F, Foxworthy P Basak, S. C. in : Topological Indices and Related Descriptors in & Eacho P, Toxicol Appl Pharmacol, 137 (1996) 75. QSAR and QSPR (Devilers, J.; Balaban, A. T., Eds.); 40 Randic M, Kleiner A F & DeAlba L M, J chem Ill/ Comput Gordon & Breach Pub!.: Amsterdam ( 1999), pp 563-593. Sci, 34 (1994) 277. Trinajstic, N.; Klein, D. J.; Randic, M. On Some Solved and 41 Second Indo-US Workshop on Mathematical Chemistry, Unsolved Problems of Chemical Graph Theory. Int. J. May 3~-June 3, 2000, Duluth, MN; J chem In/ CompL/l Sci QuantuTll Chem: Quantum Chem. Symp. 1986,20,699-742. 41 (2001) pp 479-701 (total of28 papers). Randic, M.; Trinajstic, N. Notes on Some Less Known Early 42 Randic M, Plavsic D & Razinger M, MATCH Commun math Contributions to Chemical Graph Theory, Croat. Chern. Comput Chem, 35 (1997) 243. Acta 1994,67, 1-35. Balaban, A. T. Chemical Graphs: Looking Back and Glimpsing Ahead. J. Chern. Illf. Compo Sci. 1995, 35, 339-350. Appendix 1 Randic, M. On characterization of Chemical Structure. J. Chern. Inf. Compo Sci. 1997,37,672-687. Introductory Graph Theory books: Randic, M. The Connectivity Index 25 Years After. J. Mol. Ore, O. Graphs and Their Uses, Random House: New York: Graphics & Modelling 2001, 20,19. ( 1963). Randic, M. Chemical Structure - What is "She." J. Chern. Educ. Wilson, J. R. Introduction to Graph Theory, Oliver and Boyd: 1992, 69,713-718. Edinburgh (1972). Balasubramanian, K. Applications of Combinatorics and Graph Bondy, J. A.; Murty, U. S. R. Graph Theory with Applications, Theory to Spectroscopy and Quantum Chemistry. Chern. MacMillan Press, Ltd: London (1976). Rev. 1985,85,599-618. Chartrand, G. Graphs as Mathematical Models, Prindle, Weber & Pogliani, L. From Molecular Connectivity Indice to Schmidt, Inc.: Boston 1977. Semiempirical Connectivity Terms: Recent Trends in West, D. B. Introduction to Graph Theory. Prentice-Hall, Upper Graph Theoretical Descriptors. Chern. Rev. 2000, 100, Saddle River, NJ , 1996. 3827-3858. Konig, D. Ein/iihrung in die Theorie der Endlich en !/lui Balaban, A. T. Solved and Unsolved Problems in Chemical Graph Unendliched Graphes, Chelsea: New York (1950). Theory, Annals Discrete Math. 1993,55, 109-126. Balaban, A. T. Chemical Graphs. Part 49: Open Problems in the Textbooks and advanced textbooks on GT: Area of Condensed Polycyclic Benzenoids: Topological Ore, O. Theory 0/ Graphs, Am. Math. Soc. Providence: RI (1962) Stereoisomers of Coronoids and Congeners. Rev. Rourn. Harary, F. Graph Theory, Addison-Wesley: Reading, MA (1969) Chim. 1988, 33,699-707. Bucvkley, F.; Harary, F. Distance in Graphs, Addison-Wesley Balaban, A. T. Challenging Problems involving Benzenoid Pub. Redwood City, CA, 1990. Polycyclics' and Related Systems. Pure Appl. Chern. 1982, Bursaker, R. G.; Saaty, T. L. Finite Graphs and Networks, 54, 1075-1096. McGraw-Hill, NY, 1965. Balaban, A. T. Is Aromaticity Outmoded? Pure Appl. Chern. Harary, F.; Palmer, E. M. Graphical Enumerations. Academic 1980,52, 1409-1492. Press, New York, NY 1973 Tutte, W. T. Graph Theory, Addison-Wesley: Reading, MA 1984 Appendix 2

For definitions of graph theoretical terminology see: We collected numerous questions that cover various topics and Essam, J. W.; Fisher, M. E. Rev. Mod. Phys. 1970,42, 271. 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 18. What is Dellaney triangulation of a map? How is it they know. We do not supply the an swers but they can be found constructed? Why may it be of interest in chemistry? by contacting the literature of CG and CGT. To assist interested 19. What is the "monster" graph? How bi g is the monster graph readers to find correct answers we li sted in the Appendix (how many vertices)? references that will help to find answers to specifi c questions. 20. How was the Wiener index defined by Harry Wiener and how is it related to the distance matrix? List of questions to test one's knowledge of Graph Theory and 2 1. What is a structural interpretation of the Wiener index? Chemical Graph Theory 22. What is the difference between )(jer & Hall's valence There is no significance to the order in which the following connectivity index and the relatively recently proposed questions li sted appear. Equally, some questions are more variable connectivity index? important and some are less important either for GT or CGT, 23. Why is phenanthrene more aromatic than anthracene? some questions are more difficult even if they may be 24. Is it true that n-alkane always have the highest BP (Boiling intermingled with easier questions. In reviewing the questi ons we point) among 'all isomers having th e same n? If yes - why advise that one keep hi s/her score by adding + I for every correct yes? [f no - why no? answer and subtracting a penalty of -2 for every incorrect answer, 25. What is the difference between a graph matrix (such as the while having zero fo r unanswered questions. Penalty has been adjacency matrix, the distance matrix) and graphical deliberately given a greater weight as a wrong answer requires matrices of graphs? first to eradicate the error and then to learn then correct answer. 26. It is well known that one can easi ly obtain the adjacency Observe that several questi ons have two and three sub-questi ons, matri x fro m a di stance matri x. Can one construct (not which ought to be considered separately and counted separately. If looking at a picture of a graph) the distance matri x from the you answer all questions correctly yo u can have as much as 100 adjacency matri x? points and proudly claim to be an expert on GT and CGT. Even a 27. Do topological indices have to have physic al meaning? If the score of 50 would indicate a fair fa mili arity with GT and CGT, answer is yes, then why? but anything below 50 poi nts to lack of familiarity with GT and 28. [s parti al ordering of interest in chemistry? Give an CGT. illustrati on if the answer is positi ve. 29. What is the difference between molecular topological indices Here is a list of questions: and molecul ar topographic indices? I. What is Clarke's theorem about? Why is this theorem so 30. What is the major result of a work of Coll atz and important? Sinogowitz? 2. What is the di ffe rence between the Characteristi c polynomi al 3 1. What is "graph Reconstruction" problem? Who and when and Matching polynomial? proposed the problem? Has the problem been solved? 3. John Pl att made two important contributions to chemical GT. 32. What is the contribution of Joshu a Lederberg (Nobel What are they? Laureate) to chemical notation? 4. Why is Petersen graph of interest in chemistry? What does it 33. What is the contribution of Vlado Prelog (Nobel Laureate) to represent in chemistry? chemi cal notation? 5. When has been published the first book on GT? In what 34. What are Ugi 's BE matrices? language? Who was the author? What was his nationality? 35. What are the uses of canonical labeling of verti ces of graphs: 6. In what year was published th e first book on Chemical GT? To solve Graph Isomorphi sm problem? To solve Graph Who was the editor? automorphism problem? Both? 7. What is a DID matrix? What structural interpretation has its 36. Who was first to publish a paper illustrating subgraphs leading eigenvalue? contributing to the construction of the characteristic 8. What is the elegant algorithm of Gordon and Davidson for? polynomial? 9. What is a theorem of John and Sachs about? 37. Who and when was the first to use term "graph" for 10. What is the difference between the "conjugated circuits" and mathematical objects known today as graphs? "circuits of conjugation"? Both terms were used in the 38. Who was first to consider enumeration of graphs? literature. 39. Who and when published the first paper on enumeration of II. Consider use of a single variable connectivity index (based chemical isomers? In what journal? In what language? on several variables) in structure-property-activity regressions. Does this then represent a simple regression or a 40. What are (vertex and edge) transitive graphs? Why are they multivariate regression analysis? of interest in chemistry? Who and when introduced first such 12. What is the major advantage of orthogonalized molecular graph in chemistry? descriptors in comparison with standard (non-orthogonal) 41. What is the size of the smallest edge (but not vertex) descriptors? transitive graph? Who constructed this graph? 13. What is the difference between graphical and graph 42. What graphs are cages labeled as cages? Is the Coxeter graph theoretical methodology? (vertex and edge transitive cubic graph on n = 28 vertices) a 14. Can a graph theoretical index characterize chirality? Gi ve an cage? example if possible or state "not possi ble". 43. What is graph center? What is graph diameter? What is girth 15. What is the difference between a graph theoretical invariant of a graph? and a topological index? 44. Who classified graph algorithms as NP (non-polynomial ) 16. Who introduced Distance matrix in GT? and P (polynomial)? 17. What are the Kronecker and the Hadamard product of 45. Is enumeration of paths of different length in a graph solved matrices? 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 Q 6: See the first edition of Konig, D. Einfiihrung in die by a non-polynomial (NP) or polynomial algorithm? Theorie der Elldlichell und Unelldliched Graphes, which 47. What is the detour matrix? Who introduced the detour matrix was reprinted by Chelsea: New York in 1950. in GT? Does it have another name? Q 7: See: Randic, M.; Kleiner, A. F.; DeAlba, L. M. J. Chem. 48. Can two non-isomorphic graphs have identical distance In! Comput. Sci. 1994, 34, 277. matrix? Can two non-isomorphic graphs have identical Q 8: See: Gordon, M. ; Davison, W. H. T. 1. Chem. Phys. 1952, detour matrix? 20,428. 49. How is Hosoya's topological index constructed? See also: Randic, M. Aromaticity of Polycyclic Conjugated 50. What is the Frobenius theorem that defined bounds on the Hydrocarbons. Chem. Rev. (in print). leading eigenvalue of a matrix? Q 9: See: John, P.; Sachs, H. Calculating the Numbers of Perfect Matchings and of Spanning Trees, Pauling's 51. What are "migrating" sextets? Orders, the Characteristic Polynomial, and the 52. Has Klaus Ruedenberg published a paper that can be viewed Eigenvectors of a Benzenoid system. Topics Curro Chem. as CGT paper? 1990, /53, 146-179. 53. Has Hans Primas published a paper that can be viewed as Q 10: See: Randic, M. Aromaticity of Polycyclic Conjugated CGT paper? Hydrocarbons. Chem. Rev. (in print). 54. Has C. A. Coulson published a paper that can be viewed as Q II: Some think "yes" (Zefirov, N. S.; Palyulin, V. A. QSAR CGT paper? for boiling points of "small" sul tides. Are the "H igh­ Has Linus Pauling (doubly Nobel Laureate) published a paper that Quality Structure-Property-Activity Regressions" the real can be viewed as CGT paper? high quality QSAR model. J. Chem. In! Compul. Sci. Has G. Wheland published a paper that can be viewed as CGT 2001,4/, 1022-1U27) but some think the correct answer is paper? "no" (see: Randic, M. Variable Molecular Descriptors and Has Rudolf Marcus (Nobel Laureate) published a paper that can High-Quality Regression Analysis. Understanding and be viewed as CGT paper? Misunderstandings. J. Chem. In! Comput. Sci. 55. What is the crossing number? What is the crossing number of (submitted). 4-D cube? Q 12: See: Randic, M. Resolution of ambiguities in structure­ 56. Are fullerene graphs planar graphs? property studies by use of orthogonal descriptors. 1. Chem. 57. What is Laplace matrix and what for it is used in GT and In! Comput. Sci. 1991, 31 , 3 11 -320; Randic, M. CGT? Orthogonal molecular descriptors. New.l. Chem. 1991 , 15, 58. Are th e row sum s of a symmetric matrix invaraints? 517-525; Randic, M. Fitting of non-linear regressions by orthogonalized power series. J. Compul. Chem. 1993, 14, Is the sum of all matrix elements of a symmetric matrix an 363-370. invariant? Q 13: For illustrations of a graphical approaches see: 59. Should a shape index depend on molecular size (number n of Randic, M. Graphical Enumeration of Conformations of vertices)? Do Kier's K2 and K3 depend on n? Chains. Int. J. Quantum Chem: Quantum Bioi. Symp. 60. Do we have enough topological indices? If yes - why yes? If 1980, 7, 187-197. no - why no? Klavzar,S.; Zigert, P.; Gutman, I., Theochem (submitted). Randic, M. Polycyclic Aromatic Hydrocarbons We list here sources for checking the answers to the 60 (submitted). questions li sted in the text rather than providing the answer in Q 14: See: Liang, c.; Mislow, K. J. Math. Chem. 1995, 18, 1- order that an effort is made to find the answer. For instance, on 24; Randic, M. ; Razinger, M. J. Chem. In! Comput. Sci. question 5 we could have li sted easily the name of the author, the 1996, 36, 429-442; Randic, M. 1. Chem. Ill! Comput. Sci. title of the book (which would reveal the language), indicate the 2001 ,41,639-649. nationality of the author, and even giving some biographical Golbraikh, A.; Bonchev, D.; Tropsha, A. J. Chem. In! details (for instance, the author of the book has committed suicide Comput. Sci. 2001, 41,147- 158. during WW II). But that would be counterproductive to our desire Schultz, H. P.; Schultz, E. B.; Schultz, T. P. 1. Chem. In! that readers do the investigative research. Hence, we give hints Comput. Sci. 1995, 35, 864-870. where the answers could be found and at a later time we plan to Q 15: See any of many review articles on topological indices, give the answers and accompanying information. which are more popular (but somewhat incorrect) name for graph theoretical invariants also referred to as Answers to question can be found in: molecular descriptors. Q I: Journal a/Graph Theory Q 16: See: Harary, F. Graph Theory, Addison-Wesley: Reading, Q 2: See: Trinajstic, N. Computing the Characteristic MA (1969) Polynomial of a Conjugated System Using the Sachs Q 17: See any textbook on Linear Algebra or MATLAB Manual. Theorem. Croat. Chem. Acta 1977,49, 593-633. Q 18: Dellaney triangulation is dual of a map which is divided in Q 3: See: Platt, J. R. in Handbuch der Physik (Fliigge, S. Ed.), regions such that all points with a region are closer to the Springer-Verlag: Berlin (1961), pp. 205-209; and: Platt, J. center of the region than to any other neighboring center. R. Prediction of isomeric differences in paraffin See any textbook on advanced geometry. properties. J. Phys. Chem. 1952,56,328-336. Q 19: See: Randic, M.; Oakland, D. O.; Klein, D. 1. J. Compul. Q 4: See: Dunitz, J. D. ; Prelog, V. Allgew. Chem. 1968, 80, Chem. 1986, 7, 35. 700; Balaban, A. T.; Farca~iu , D.; Banicii, R. Rev. Roum. Zivkovic, T. Croat. Chem. Acta Chim. 1966, II, 1205. Q 20: See: Wiener, H. Prediction of isomeric differences In Q 5: See: Graph Theory 1736 - 1936. paraffin properties. J. Am. Chem. Soc. 1947,69, 17-20. RANDle: CHEMICAL GRAPH THEORY 121 7

See: Hosoya, H. Topological index. A newly proposed quantity Q 35: See: Read, R. c.; Corneil, D. G. The Graph Isomorphism characterizing the topo logical nature of structural isomers Di sease, l . Graph Theory 1977, 1,339. of saturated hydrocarbons. Bull. Chem. Soc. l apan, 1971, Q 36: See: Coulson, C. A. Proc. Cambridge Phil. Soc. 1950, 46, 44,2332-2339. 202. Q 21 See: Randic, M.; Zupan. 1. On Structural Interpretation of Q 37: See: Rouvray, D. H., l . Mol. Stmct. 1989, 185, I ;Bell, E. Topological Indices, In : Proc. Wiener Melllorial T. Men of Mathelllatics; See also an article entitled Conference. Athens 2001 (R. B King and D. H. Rouvray, Chemistry and Algebra by Sylvester, 1. J. Nature, 1877- Eds.); Randic, M.; Zupan, J. On interpretation of well ­ 1878, XVll. pp. 284, 309. known topological indices. 1. Chen!. In! Comput. Sci. Q 38: See a chapter on graph enumeration in: Biggs, R; Lloyd, 2001 ,41,550-560. E. K.; Wilson, R. J. Graph Theory 1736 - 1936, Q 22:See: The Connectivity Index 25 Years after (J. Molecular Clarendon Press: Oxford, 1976. Graphics alld Modelling, 2001,20, 19-35). Q 39: See: Balaban, A. T. Enumi!fatioll of Isomers. In Chemical Q 23: See: Randic, M. Aromaticity of Polycyclic Conjugated graph theory. Abacus Press, Gordon & Breach: New Hydrocarbons, Chelll. Rev. (in press). Also: Randic, M. l . York, 1990, p. 177-234. Am. Chefl/. Soc. 1976, Randic M.; Trinajstic, N. On some Less Known Graph Theory Q24: See Randic M; Wilkins. C. L. On graph theoretical basis for Contributions, Croat. Chell!. Acta 1994,67, 1-35. ordering of structures. Chelll . Phys. U fl . 1979, 63, 332- Q 40: See: Trinajstic, N. Chemical graph theory, (2nd ed.) CRC 336: Randic M; Wilkins, C. L. Graph th eoretical ordering Press: Boca Raton, FI 1992. of structu res as a basis for systematic searches for Q 41: See: Pi sanski, T.; Randic, M. Bridges between Geometry regularity in molecul ar data. l. Phys. Chem. 1979, 83, and Graph Theory; in : Geometry at Works (Papers in 1525- 1540. Applied Geometry), Gorini C. A., Editor, MAA Q 25: See: Randic, M .; Plavsic, D.; Razinger, M. Double (Mathemati cal Association of America) Notes Number 53, In vari ants. MATCH-Comlllun. Math. CO llljJLIl. Chem. Washington, D. c.: 2000 or: 1997,35,243-259. Bondy, J. A.; Murty, U. S. R. Graph Theory with Applications, Q 26: See: Kunz, M. An equi valence relation between distance MacMillan Press, Ltd: London (1976). and coordinate matrices. MATCH - Comlllull. Math. Q 42: See: Balaban, A. T. 1. Combinatorial Theory, Ser. B Comput. Chen!. 1995,32, 193-204. 1972, 12, I ; Balaban, A. T. Rev. Roum. Math. Pures Appl. Q 27: See: Randic, M.; Zupan, 1. On the structural interpretation 1973, 18, 1033; Also: Wong, P. K Cages - a survey. l . of topological indices, in : Topology ill Chemistry- Discrete Graph Th eory. 1982, 6, 1-22. Mathematics of Molecules (Rouvray, D. H.; King, R. B., Eds.), Horwood Pub!.: Chichester, England, 2002, pp. Q 43: See: Harary. F. Graph Theory, Addison-Wesley: Reading, 249-29 1. MA ( 1969);Bonchev, D.; Balaban, A. T.; Randic, M. Int. Estrada, E. The Physicochemical Interpretatio n of Molecul ar l . QuallIum Chem. 1981, 19,61. Connectivity Indices, l . Phys. Chell!. (in press) Q 44: See: Miller, R. E.; Thatcher, J. W. Complexity of Q 28: See: Kl ein , D. 1. Prolegomenoll on Partial Orderings in Computer Computations. Pl enum Press, 1972, pp. 85-103. Chemistry, MATCH - Commun. Math. Compllt. Chem. Q 45: See: Miller, R. E.; Thatcher, J. W . Complexity of 2000,42, 7; Randic, M. Vracko, M.; Novic, M. Basak, S. Computer Computations. Plenum Press, 1972, pp. 85- 103. C. MATCH - Com/1/un. Math. Comput. Chern. 2000, 42. Q 46: Harary, F. Graph Theory, Addison-Wesley: Reading, MA 18 1. ( 1969). Q 29: See: Randic, M.; Razinger, M. Molecular Topographical Q 47: See: Trinajstic, N.; Nikoli c, S.; Lucic, B.; Amic, D.; Indices. 1. Chem. In! Comput. Sci. 1995, 35, 140- 147; Mihalic, Z. The Detoru Matri x in Chemistry. 1. C/leln. In! Also: Estrada E. Characterization of 3D Molecular Comput. Sci. 1997, 37, 631-630;Nikolic, S.; Trinajstic, N.; Structure. Chem. Phys. Lefl . 2000, 319, 7 13-718. luric, A.; Mihalic, Z. The Detoru Matrix and the Detour Q 30: See: Cvetkovic, D. M. ; Doob, M. ; Sachs, H. Spectra of Index of Weighted Graphs. Croat. Chem. Acta 1996, 69, Graphs, 3rd ed. Huthig GmbH - 1. A. Barth Verlag, 1577-1591;Lukovits, I. The Detour Index. Croat. Chem. Heidelberg, 1995; Acta 1996, 69, 873-882; Lukovits, I.; Razinger, M. On the Collatz, L. ; Sinogowitz, U. Abh. Math. Sem. Ulliv. Calculation of the Detour Index, l . Chem. In! Comput. Hamburg 1957, 21, 63. Sci. 1997, 37, 283-286.; Amic, D. ; Trinajstic, N. On the Q 3 1: See: O'Neil, P. V. Ulam's Conjecture and Graph Detour Matrix. Croat. Chem. Acta 1995,68,53-62. Reconstructi on. Amer. Math. Montly, 1970, 77, 35-43. Q 48: See: Randic, M.; DeAlba, L. M. ; Harris, F. E. Graphs Q 32: See: Lederberg, 1.; Sutherland, G. 1.; Buchanana, B. G.; with the same Detour Matrix. Croat. Chem. Acta 1998, 71 , Feigenbaum, E. E.; Robertson, A. V.; Duffield, A. M.; 53. Dj erassi, C. Application of Artificial Intellige nce for Q 49: See: Hosoya, H. Topo logical index. A newly proposed Chemical Inferences. I. Ther Number of Possible Organic quantity characterizing the topological nature of structural Compounds. Acyclic Structures Containing C, H, 0, and isomers of saturated hydrocarbons. Bull. Chem. Soc. lpn. N. l . Am. Chern. Soc. 1969,91,2973-2976 .. 1971 , 44,2332-2239. Q 33: See: Nobel Laureates in Chemistry 1901 - 1992 (Laylin Q 50: See a textbook on Linear Algebra K lames, Ed.) Amer. Chem. Soc. Washington, D. c., Q 51: Clar, E. The Aromatic Sextet J. Wiley & Sons: London 1993; Cahn, R. S.; Ingold, C. K; Prelog, V. Angew. Chem. 1972 Int. Engl. Ed. 1966, 5, 385. Q 52: See: Ruedenberg, K l . Chem. Phys. 1954, 22, 1878; Q 34: See: Dugundji, J.; Ugi, I. Topics Curro Chem.1973, 39, Ham, N. S.; Ruedenberg, K l . Chem. Phys. 1958, 29, 19. 1215. 1218 INDIAN J CHEM, SEC A, JUNE 2003

Q 53: See: Glinthard. H. H.; Primas, G. Helv. Chill/. Acta 1956, Hahn. G.: Sabidussi. G. (Eds.) Graph Symmetry: 39, 1645. Algebraic Methods alld Applications, (NATO ASI seri es, Q 54: . See: Coulson, C. A. Proc. Call/bridge Phil. Soc. 1950,46, Series C, Mathematical and Physical sciences, vol. 497). 202: Pauling, L. 1. Chem. Phys. 1933, I, 280; Wheland, G. Dordrecht; Boston; London: Kluwer, 1997, pp. 225-270. W. 1. Chem. Phys. 1933, 3, 356: Marcus, R. A. 1. Chem. Q58: True only for regular graphs in which case the row sums are Phys. 1965, 43,2643. constant a,nd represent trivial invariant (of little intereest). Q 55: See: Harary, F. Graph Theory, Addison-Wesley: Reading, Q 59: See: Randic, M. Novel Shape Descriptors for Molecular MA ( 1969). Graphs. 1. Chem. Inf. Call/put. Sci. 2001, 41, 607-613. Q 56: All polyhedral graphs are planar Q 60: The answer is "yes" if you want to freeze the development Q 57 : See, for instance, Mohar, B. Laplace eigenvalues of graphs of chemical graph theory as a scientific di scipline; if you - a survey. Discrete Math. 1992, 109, 171-183; Mohar, B. think that science is not static, the answer is an emphatic Some applications of Laplace eigenvalues of graphs. "no."