Acta Chim. Slov. 2002, 49, 483–496. 483 ON GENERALIZATION OF WIENER INDEX FOR CYCLIC STRUCTURES Milan Randić National Institute of Chemistry, Ljubljana, Slovenia; e-mail: milan.randic@ki.si and Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311, USA Home FAX: 515 292 8629 Received 15-11-2001 Abstract We have critically examined a particular generalization of Wiener index to cyclic structures known as Szeged index. Limited ability of Szeged index to give fair correlation of boiling points of cycloalkanes was traced to a deficiency in its definition which leads to great variation in index values between odd member cyclic systems and even member cyclic systems, both of which show similar boiling points. A remedy is suggested for construction of Wiener index more suited for cyclic structures. Mathematical elegance is not enough for good topological index. Introduction Mathematical descriptors of molecular structure, such as various topological indices,1 have been widely used in structure-property-activity studies. This includes the multiple regression analysis (MRA), the principal component analysis (PCA),2 the pattern recognition,3 the artificial neural networks (ANN),4 optimization of lead compound and combinatorial optimization of the lead compound,5 search for pharmacophore,6, 7 and the similarity-dissimilarity studies. Apparently there is no lack of initiative and imagination in proposing novel topological indices, but there appears to be lack of efforts to demonstrate of their specificity and lack of efforts in clarification of their structural meaning. By specificity we understand here useful characterization of one of molecular properties by the novel index when used alone, or in combination with other molecular descriptors. By structural meaning of an index we understand interpretation of an index in terms of elementary structural concepts such as atoms, bonds, or in terms of paths and walks of longer length. Only recently8,9 such an interpretation was offered for several well known indices, including the connectivity index,10 the Wiener index,11 and the Hosoya Z topological index.12 Lists of desirable M. Randić: On generalization of Wiener index for cyclic structures 484 Acta Chim. Slov. 2002, 49, 483–496. qualities for novel topological indices have been proposed,13,14 but have also been mostly ignored or overlooked. In this article we will focus attention on generalization of the Wiener index for use in cyclic structures. As is well known, Wiener11 defined his W index only for acyclic graphs. Attempts have been reported to generalize Wiener index to cyclic graphs. An index proposed by Gutman,15 known as Szeged index, received considerable attention.16-23 It has attractive mathematical characteristics, but does it offer useful characterization of cyclic structures? In this paper we have critically re-examined the Szeged index and found it deficient as a molecular descriptor for structure-property-activity studies, despite it's elegant and attractive mathematical definition. We will present its "structural" deficiency and will show how the observed deficiency can be corrected. This resulted in a novel index, revised Wiener index RW, which as will be seen shows better descriptor for structure – property relationship for cyclic molecules. One may say that there are essentially two reasons for considering novel or alternative molecular descriptors: (1) Novel descriptors may lead to better regression analysis, and (2) Novel or alternative descriptors may be computationally simple, which is important when screening combinatorial libraries which may have 100,000 structures or more. To this we would like to add an additional consideration: (3) Novel or alternative descriptors may offer a simpler structural interpretation. There are hundreds of available mathematical descriptors for molecular graphs, many of which can be computed by programs such as MOLCONN,24 POLLY,25 CODESSA.26 For a recent compilation of large number of topological indices readers are directed to a book of Todeschini and Consinni.27 In Table 1 we have collected several recently introduced topological indices still based on rather simple structural components, merely to illustrate structural diversity of topological indices. Among those listed is also the revised Szeged index, and the revised "higher order" Szeged indices (analogous to the hyper-Wiener index of acyclic structures). In this paper we will closely examine the revised Szeged index, but as will be apparent similar modifications considered here for Szeged index apply also to the "higher order" indices which use paths of different length instead or considering only bonds (paths of length one). Finally, we should add that most of distance related M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 485 indices can be associated with matrices, which are either sparse (when considering only adjacent atoms) or dense (when considering paths of different length). A number of novel graph matrices have been introduced in recent years that include for instance the Wiener matrix,28 the Hosoya Z matrix,29 the Restricted Random Walk matrix,30 the Distance/Distance matrix,31 the Resistance-Distance matrix,32 the Detour matrix,33 two kinds of Path matrices,34,35 etc. In this way recently a dozen novel atomic descriptors were generated. We should add that the first such matrices beyond the traditional adjacency and distance matrix are the Expanded Wiener matrix of Tratch, Stankevich and Zefirov,36 and the electrotopological matrix of Kier and Hall.37 Table 1. Several novel topological indices based on use of different structural factors index structural factors involved Shape index w/p given by quotient of paths and walks of a same length Double invariants matrix elements given by subgraphs; numerical values by subgraph invariants Composite X'/X descriptors index is given by quotient of an invariant X and the same invariant when one bond at a time is excluded from graph Variable connectivity Parameters x, y, z, . . ., which characterize different kinds of atoms, are optimized during regression process Chirality index Derived from invariants obtained by circumscribing planar molecules in opposite direction Generalized (acyclic) Wiener Multiply the number of atoms on each side of a path for every pair of vertices Generalized (cyclic) Wiener Multiply the number of atoms on each side of an edge closer to each end atoms, add 1/2 of atoms at equal distance Generalized (cyclic) Wiener Multiply the number of atoms on each side of a path closer to each end atoms for every pair of vertices, add 1/2 of atoms at equal distance M. Randić: On generalization of Wiener index for cyclic structures 486 Acta Chim. Slov. 2002, 49, 483–496. Szeged index Wiener,11 has defined his W index as a bond additive index in which each bond makes a contribution (m x n), where m and n are the number of atoms on each side of a bond considered. This definition allows a generalization28,38,39 in which all pair of atoms makes contribution, not only adjacent atoms. One simply replaces "(m x n), where m and n are the number of atoms on each side of a bond considered" by "(m x n), where m and n are the number of atoms on each side of a path considered." In this way a matrix is obtained, the so-called Wiener matrix,38 from which the hyper-Wiener index38,39 was extracted. This generalization, however, has not considered cyclic structures, which constitute a large portion of chemistry. Gutman has considered generalization of W to cyclic structures and came with mathematically elegant definition of the Wiener index for cyclic structures that bears some analogy to already mentioned design of hyper-Wiener index. Gutman has replaced "(m x n), where m and n are the number of atoms on each side of a bond considered" in the definition of Wiener by "(m x n), where m and n are the number of atoms closer to atoms on each side of a bond considered." Clearly in cyclic atoms one can not speak of "of atoms on each side of a bond" because bond does not separate cyclic structure in two parts. However, in both cases, acyclic and cyclic compounds, one can speak of "of atoms closer to atoms on each side of a bond." What is wrong with Szeged index Szeged index has an elegant mathematical formulation but from its limited applications it appears not to offer useful correlation. First observe that of some dozen papers on Szeged index most of these are concerned with mathematical properties of this index, its relationship to other indices, and its evaluation for special classes of molecules rather then demonstrating its use in chemical applications. Very few of these contributions mention any correlation with any of physicochemical properties examined by other indices. It is nothing wrong with studying mathematical properties of graph invariants, but unless it first is demonstrated that a particular invariant has some use in the structure-property-activity relationships such mathematical studies are premature, to say the least. M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 487 The only paper that we found in which Szeged index is tested in QSPR (quantitative structure-property relationship) is paper by Diudea40 in which he examined use of Szeged index (and several other distance-based indices) on correlation with the boiling points (BP) of 45 cycloalkanes. When in a simple regression one uses N, the number of carbon atoms in a molecule, one obtains regression shown in Fig. 1 characterized by the regression coefficient r = 0.969, the standard error s = 10.9; and the Fisher ratio F = 664. 300 200 B P 100 0 H--------------1--------------1--------------1--------------1-------------1--------------1--------------1--------------1 3456789 10 11 N Fig. 1. The regression of the boiling points of cycloalkanes against N Clearly all isomers (having the same N) are grouped together displaying limitations of N as molecular descriptor to differentiate isomeric variations. Any index that does not do better than N, which is only a measure of molecular size, can be viewed, at least for the particular application, as useless. The simple regression of the boiling points of cyclic alkanes based on the Szeged index and shown in Fig. 2 is characterized by the following statistical parameters: r = 0.919, s = 17.7; and F = 225 -hence, useless, being worse than the regression based on N, the number of carbon atoms in a structure. M. Randić: On generalization of Wiener index for cyclic structures 488 Acta Chim. Slov. 2002, 49, 483–496. 300 200 100 0 T 100 200 SZEGED Fig. 2 Regression of the boiling points of cycloalkanes against Szeged index Table 2. The boiling points and Szeged indices for N = 8 cycloalkanes isomers 300 isomer BP Szeged isomer BP Szeged Even rings Odd rings 11MC6 119.5 104 112MC5 114 67 12MC6 123.4 106 113MC5 105 71 13MC6 124.5 108 123MC5 115 70 14MC6 120 110 12MEC5 124 72 EC6 131.8 109 13MEC5 121 76 C8 146 128 PC5 131 78 IPC5 126.4 73 MC7 134 88 Average 127.5 110.8 Average 121.3 74.4 Average* 123.8 107.4 Average* 119.5 72.4 0 M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 489 Why is that Wiener index (in combination with polarity index P) has produced outstanding regressions for several physico-chemical properties of alkanes and other compounds as described over 50 years ago by Wiener while its generalization to cyclic compounds is not producing good correlation at all? The problem is, as we have found, with the proposed generalization of the Wiener index as offered by Gutman. In Table 2, we have listed for cycloalkanes having N = 8 their boiling points and their Sz indices separating compounds having even and those having odd member rings. As we see from Table 2 on average the BP of cycloalkanes having even member ring and those having odd member ring do not differ greatly. On the other hand the average the Szeged numbers for the two kind of cycloalkanes show an excessive difference in their relative magnitude. This is even more pronounced in the last row of Table 2 in which we have confined attention only to five and six member ring compounds. The average BP of the two groups differ by about 4 °C (about 3%), while the Szeged index of the two groups differ by 35 units (more than 30%). Clearly the Szeged index cannot be successive in correlating properties which do not depend dramatically on the parity of ring size. Improved Wiener index for cyclic structures Now that we pointed to inadequacy of Szeged index we will outline a design of an improved or revised Wiener index (RW) for cyclic structures. From Table 2 it can be inferred that problem is with the Szeged index for compounds having odd rings, the magnitude of which are too small, and not with the Szeged index of compounds with even member rings. This is based on the fact that Wiener index for acyclic compounds having the same number of CC bonds (e. g. n-nonane) is around 100 and not around 75. One can immediately recognize that reduced size of Szeged index of compounds having odd rings is due to neglect of contributions from atoms (and bonds) which are at the same distance from both atoms forming a bond. Hence, we have to augment Szeged index with contributions from atoms not considered in the definition of Sz index as proposed by Gutman. A simple remedy to deficiency of Sz is to divide equally the count atoms at the same distance from atoms at both ends of a bond. This is illustrated in Fig. 3 on 1-methyl-2-ethylcyclopentane (12MEC5). As we see, instead of having M. Randić: On generalization of Wiener index for cyclic structures 490 Acta Chim. Slov. 2002, 49, 483–496. Sz = 72 we obtain RW = 100, which is close to Szeged indices for isomers having even member ring. In this way we obtain the revised Szeged indices of Table 3 for cycloalkanes having odd ring. 6 4_____3 C1 - C2 = 3.5 x 4.5 C2 - C1 - C5 = 5.5 x 2.5 C3 - C1 - C6 = 7 x 1 C4 - C2 - C3 = 5.5 x 2.5 C7 - Fig. 3 Calculation of revised Wiener index RW for 1-methyl-2-ethylcyclopentane Table 3. Szeged index and the Revised Wiener index RW for cyclic structures M = methyl; E = ethyl; P = propyl; IP = isopropyl Szeged RW Szeged RW 11MC3 15 24.75 113MC5 71 99 EC3 17 26.75 123MC5 70 98.5 C5 20 31.25 12MEC5 72 100 112MC3 26 40.5 13MEC5 76 104 123MC5 27 42 PC5 78 105 MC5 33 49 IPC5 73 100 11MC5 48 69.25 MC7 88 117.5 12MC5 49 70.75 1123MC5 93 128.75 13MC5 51 72.75 EC7 121 157.75 C7 63 85.75 C9 144 182.25 112MC5 67 95 PC7 163 207.5 C7 = 6 x 2 C4 = 5 x 3 C5 = 3.5 x 4.5 C8 = 7 x 1 M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 491 300 S Z E G E D 200 100 0 T 100 200 300 REVISED Fig. 4 Correlation between Szeged index and Revised Wiener index for cyclic structures In Fig. 4 we show a correlation between the Szeged index as defined by Gutman and others and the revised Szeged index RW for the 45 cycloalkanes considered by Diudea in his paper on Cluj matrices. From the figure one immediately sees a "double" line pattern, one given by equation y = x, and the other below it showing some minor departures from a strict line. The upper line belongs to cyclic alkanes having even ring, in which case the Szeged index and RW are identical. Compounds having even member ring need no correction because in such systems always an atom is closer to one end of any CC bond. The small oscillations in the lower line are caused by different number of "ignored" atoms in computation of the Szeged index. For example, for 1MC7 only one carbon (of the methyl group) has been "ignored" in calculating the contribution for a CC bond on the opposite site of methyl group in the 7-member ring. In contrast in PMC5 four carbon atoms have been "ignored" in calculating the contribution for a CC bond on the opposite site of the propyl group in 5-member ring. 0 M. Randić: On generalization of Wiener index for cyclic structures 492 Acta Chim. Slov. 2002, 49, 483–496. Correlation of BP for cycloalkanes with Revised Wiener index RW In Fig. 5 we show correlation between BP of cyclkoalkanes and RW, the revised Wiener index for cyclic structures. 300 200 "" B P 100 "' 0 0 100 200 300 REVISED Fig. 5 Regression of the boiling points of cycloalkanes against RW index A comparison of Fig. 5 with Fig. 2 immediately shows that regression based on the revised Wiener index RW is much better. In Table 4 we have summarized statistical information on the regression using Sz and RW. As we see a quadratic regression gives for the standard error 8.3 °C as compared with 14.3 °C. This seems an impressive improvement of the regression. While this may still not be satisfactory regression from the point of view of structure-property correlation as a simple regression based on a single descriptor it certainly represents a satisfactory preliminary result. For example, recently41 a comparison is made between a correlation of the boiling points of cycloalkanes using the connectivity index10 and the variable connectivity index.42-49 In both cases the standard error was close to 3 °C , showing that bond additive descriptors can offer satisfactory correlation with boiling points in the case of cycloalkanes. M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 493 Table 4. The statistical data for regression of BP of 45 cycloalkanes using the Sz and RW. Included is statistical data for regression using N (illustrated in Fig. 1) Descriptor r s F N 0.9667 11.4 614 Szeged index 0.9207 17.4 239 Linear 0.9480 14.3 186 Quadratic Revised Wiener 0.9635 11.9 556 Linear 0.9828 8.3 596 Quadratic The Wiener index and the Szeged index, as well as here considered their modifications, even though can be viewed also as bond additive molecular descriptors, have totally different structural origin, being based on distance metrics. It is thus of considerable interest to see how these distance-related indices behave. Szeged index, which represents a particular generalization of the Wiener index to cyclic structures, failed to produce even rough correlation. However, this apparently has not been either recognized or reported in the literature to warn potential users. Thus we here want to draw attention of those interested in topological indices that clearly "something is rotten in the state of Denmark," that is in "proposed extention of the Wiener index to cyclic structures." While there is no doubt that the Szeged index has mathematical elegance, mathematical elegance alone is not enough for an index to qualify as useful topological index! M. Randić: On generalization of Wiener index for cyclic structures 494 Acta Chim. Slov. 2002, 49, 483–496. Table 5. Computed BP and associated residuals for 45 cycloalkanes using RW indices BP calc Residual BP calc Residual C4 28.5 -15.4 11MC6 120.8 -1.3 11MC3 39.3 -18.3 12MC6 122.5 0.9 EC3 41.7 -5.8 13MC6 124.1 0.3 MC4 43.2 -2.6 14MC6 125.8 -5.8 C5 47.0 2.3 EC6 125.0 6.8 112MC3 57.8 -1.3 MC7 131.8 2.2 123MC3 59.5 6.5 C8 139.8 6.2 EC4 62.6 7.8 1123MC5 140.3 -7.6 MC5 67.3 4.5 113MC6 148.2 -11.6 C6 72.8 7.9 124MC6 150.9 -14.9 PC4 87.4 22.6 135MC6 150.9 -12.4 11MC5 88.6 0.3 12MEC6 149.6 1.4 12MC5 90.1 1.8 13MEC6 152.2 -3.2 13MC5 92.1 -0.4 PC6 153.5 0.5 MC6 97.3 3.6 IPC6 149.6 -3.6 C7 104.6 12.4 EC7 159.5 4.0 112MC5 113.0 1.0 C9 172.7 -2.7 113MC5 116.5 -11.5 12MIPC6 171.6 -0.6 123MC5 116.1 -1.1 13MIPC6 174.5 -7.0 12MEC5 117.4 6.6 13EC6 177.2 -6.7 13MEC5 120.8 0.2 PC7 183.4 0.1 PC5 121.6 9.4 C10 194.7 15.3 IPC5 117.4 9.0 For a completeness of the presentation in Table 5 we show computed BP and associated residuals for 45 cycloalkanes using RW indices. As we can see four cycloalkanes are associated with the largest positive and the largest negative residuals. When these four compounds (#1, #2, #11, and #45) are removed we obtain respectable single variable regression characterized by the regression coefficient r = 0.9877, the standard error s = 6.0 °C , and Fisher ratio F = 760. We should mention that regression using Sz index has much larger scatter of points that it does not allow one to identify potential outliers, thus one is stuck with the standard error of 14.3 °C , much too large to be of use in discussion of relative boiling points of cycloalkanes. M. Randić: On generalization of Wiener index for cyclic structures Acta Chim. Slov. 2002, 49, 483–496. 495 Concluding Remarks Now that we pointed to inadequacy of Szeged index and have outlined a design of Revised Wiener index for cyclic structures we hope that researches in the field will recognize the importance of "use" of topological indices and the study of their limitations, rather than merely "studying their mathematical properties," which can only be of interest if topological index has found use. In the past there has been too much emphasis in some circles on novelty of indices, as if a novelty is guarantee of usefulness of such indices in chemical applications. The present extension of the Wiener index to cyclic structures applies equally to construction of matrices associated with distances in cyclic systems. In a follow up paper50 we have re-examined extension of here presented generalization of the Wiener index to the higher order Wiener indices based on considerations of paths of longer length and atoms at equal distance from such paths. This similarly leads to a revision of previously proposed indices in which atoms and bonds at equal distance from the end points of paths have been ignored, in our view incorrectly ignored. References and Notes 1. M. Randic, Topological Indices. Encyclopedia of Computational Chemistry (P. von Rague Schleyer, Editor-in-Chief, London: Wiley, 1998, pp.3018-3032. 2. H. Hotelling J. Educ. Psychol. 1933, 24, 8417 & 489. 3. T. Okuyama, Y. Miyashita, S. Kanaya, H. Katsumi, S. I. Sasaki, and M. Randic J. Comput. Chem. 1988, 9, 636. 4. J. Zupan, J. Gasteiger, Neural Networks for Chemists; VCH: Weinheim, 1993. 5. G. Grassy, B. Calas, A. Yasri, R. Lahana, J. Woo, S. Iyer, M. Kaczorek, R. Floc'h, R. Buelov Nature Biotechnology, 1997, 70, 819. 6. M. Randic Molecular Basis of Cancer, Part A: Macromolecular Structure, Carcinogens, and Oncogens, (R. Rein, Ed.), Alan Liss Publ., 1985, pp. 309-318. 7. M. Randic, B. Jerman-Blazic, D. H. Rouvray, P. G. Seybold, S. C. Grossman Int. J. Quantum Chem: Quantum Biol. Symp. 1987, 14, 245. 8. M. Randic, J. Zupan J. Chem. Inf. Comput. Sci. 2001, 41, 550. 9. M. Randic, J. Zupan, On Structural Interpretation of Topological Indices, in: Proc. Wiener Memorial Conference, Athens 2001 (R. B King and D. H. Rouvray, Eds.) 10. M. Randic J. Am. Chem. Soc. 1975, 97, 6609. 11. H. Wiener J. Am. Chem. Soc. 1947, 69, 17. 12. H. Hosoya Bull. Chem. Soc. Jpn. 1971, 44, 2332. 13. M. Randic J.Math. Chem. 1991, 7, 155. 14. A. T. Balaban J. Mol. Struct. (Theochem), 1988, 165, 243. 15. I. Gutman Graph Theory Notes, 1994, 27, 9. 16. I. Lukovits, W. Linert J. Chem. Inf. Comput. Sci. 1994, 34, 899. 17. P. V. Khadikar, N. V. Deshpande, P. P. Kale, A. A. Dobrynin, I. Gutman, G. Domotor J. Chem. Inf. Comput. Sci. 1995, 35, 547. 18. I. Gutman S. Klavzar J. Chem. Inf. Comput. Sci. 1995, 35, 1011. 19. D. J. Klein. I. Lukovits, I. Gutman J. Chem. Inf. Comput. Sci. 1995, 35, 50. M. Randić: On generalization of Wiener index for cyclic structures 496 Acta Chim. Slov. 2002, 49, 483–496. 20. M. V. Diudea, O. Minailiuc, G. Katona, I. Gutman MATCH, 1997, 35, 000. 21. J. Zerovnik Croat. Chem. Acta, 1996, 69, 837. 22. A. A. Dobrynin Croat. Chem. Acta, 1997, 70, 819. 23. I. Lukovits J. Chem. Inf. Comput. Sci. 1998, 38, 125. 24. L. H. Hall MOLCONN-X; Hall Associates Consulting, Quincy: MA, 1991 25. Basak, S. C. POLY; Natural Resources Research Institute, Duluth, University of Minnesota: Duluth, MN, 1988. 26. Katritzky, A. R.; Lobanov, V.; Karelson, M. CODESSA (Comprehensive Descriptors for Structural and Statistical Analysis); University of Florida: Gainesville, FL, 1994. 27. R. Todeschini, V. Consonni Handbook of Molecular Descriptors (Methods and Principles in Medicinal Chemistry, vol. 11, (R. Mannhold, H. Kubinyi, H. Timmerman, Eds.), Wiley-VCH. 28. M. Randic Chem. Phys. Lett. 1993, 221, 478. 29. M. Randic Croat. Chem. Acta 1994, 67, 415. 30. M. Randic Theor. Chim. Acta 1995, 92, 97. 31. M. Randic, A. F. Kleiner, L. M. DeAlba J. Chem. Inf. Comput. Sci. 1994, 34, 277. 32. D. J. Klein, M. Randic J. Math. Chem. 1993, 12, 81. 33. D. Amic, N. Trinajstic Croat. Chem. Acta 1995, 68, 53. 34. M. Randic, D. Plavsic, M. Razinger MATCH 1997, 35, 243. 35. O. Ivanciuc, A. T. Balaban MATCH 1994, 30, 141. 36. S. S. Tratch, M. I. Stankevitch, N. S. Zefirov J. Comput. Chem. 1990, 11, 899. 37. L. B. Kier, L. H. Hall Molecular Structure Description: The Electrotopological State: Academic Press (1999). 38. M. Randic, X. Guo, T. Oxley, H. Krishnapriyan J. Chem. Inf. Comput. Chem. 1993, 33, 709. 39. M. Randic, X. Guo, T. Oxley, H. Krishnapriyan, L. Naylor J. Chem. Inf. Comput. Chem. 1994, 34, 361. 40. M. Diudea J. Chem. Inf. Comput. Sci. 1997, 37, 300. 41. M. Randic, D. Plavsic, N. Lers J. Chem. Inf. Comput. Sci. 2001, 41, 657. 42. M. Randic Chemom. Intell. Lab. Syst. 1991, 10, 213 43. M. Randic J. Comput. Chem. 1991, 12, 970. 44. M. Randic, J. Cz, Dobrowolski Int. J. Quantum Chem: Quantum Biol. Symp. 1998, 70, 1209. 45. M. Randic, D. Mills, S. C. Basak Int. J. Quantum Che:. 2000, 80, 1199. 46. M. Randic, S. C. Basak J. Chem. Inf. Comput. Sci. 2000, 40, 899. 47. M. Randic, S. C. Basak J. Chem. Inf. Comput. Sci. 2001, 41, 619. 48. M. Randic, M. Pompe J. Chem. Inf. Comput. Sci. 2001, 41, 631. 49. M. Randic New J. Chem. 2000, 24, 165. 50. M. Randic, M. Novic, M. Vracko, J. Zupan J. Chem. Inf. Comput. Sci. (to be submitted) Povzetek Kritično smo pregledali posebno posplošitev Wienerjevega indeksa za ciklične strukture, imenovano Szeged index. Ugotovili smo, da je izvor omejene zmožnosti Szegedovega indexa, da bi izkazal dobro korelacijo z vrelišči cikloalkanov, v njegovi pomanjkljivi definiciji. Posledica take definicije je velika variacija vrednosti indeksa, izračunanega za ciklične strukture, katerih obroči imajo liho oziroma sodo število atomov, njihova vrelišča pa so podobna. Predlagamo popravek, ki nudi bolj primerno rešitev za gradnjo Wienerjevega indeksa za ciklične strukture. M. Randić: On generalization of Wiener index for cyclic structures