Acta Chim. Slov. 2000, 47, 489-506. 489 GRAPH DISSECTION REVISITED. APPLICATION TO SMALLER ALIKANES Milan Randić,a, Xiaofeng Guo,a, c and Paula Calkinsa a Department of Mathematics and Computer Science, Drake University, Des Moines, Iowa 50311, USA; National Institute of Chemistry, Hajdrihova 19, P. O. Box 3430, Ljubljana, Slovenia c Institute of Mathematics and Physics, Xinjiang University, Wulumuqi Xinjiang 830046, P. R. China Received 02-11-1998 Abstract Dissection of graphs was outlined twenty years ago as a scheme to arrive at an integer characterization of graphs. The approach, which raises some mathematical questions as well as it offers novel structural descriptors of potential interest in structure-property studies, was apparently overlooked. In this contribution we have re-examined graph dissection, we corrected some errors in the earlier publications, and illustrate use of novel descriptors derived through the dissection process in structure-property correlation for selected properties of octane isomers. Introduction With hundreds of topological indices one would expect that most molecules could be well characterized and their physicochemical properties correlated with the available descriptors. However, a number of properties even of alkanes, which are the simplest structures to consider in view that they lack the complications caused by the presence of heteroatoms, still can not be well described with available descriptors. For example, this is the case with critical temperature, critical pressure and critical volume. In addition the steric parameters of octanes also cannot be well correlated with most of the available molecular topological indices. Hence, the search for novel molecular descriptors continues. Recently a number of novel molecular descriptors were outlined. Let us mention few: The novel shape descriptors represented as the ratio of the count of paths and walks of the same length,1, 2 the novel chirality index for planar structures (e. g., planar benzenoid hydrocarbons),3 a "missing" descriptor p3* which count the paths of length three for heteroatoms.4 We would like to add to this list the descriptors a, b which count the atom and the bond components in a successive dissection of graphs, outlined twenty M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 490 Acta Chim. Slov. 2000, 47, 489-506. years ago by one of the present authors. , As we will see the novel parameters a, and b, which are integers, are on one side very sensitive to molecular branching and on the other hand appears to be unique, at least for smaller trees. Hence, their mathematical properties may be of interest, particularly if the high discrimination among isomers is shown to hold for larger structures. Here, we will be primarily interested in use of the novel descriptors a, b in structure-property relation, which is the sole and the ultimate criterion for a justification of introducing novel topological indices. Dissection of Graphs Dissection of graphs was defined as a stepwise process in which one removes one vertex at a time from the graph considered and continues to do so on all subgraphs, which are neither isolated vertices nor isolated bonds. Formally we have Definition: Dissection (a, b) of a graph G is collection of subgraphs A and B obtained by erasing one vertex at a time from a graph G and all its so obtained subgraphs G* generated from G which neither represent isolated vertices A nor isolated bonds B. In Fig. 1 we illustrate the steps involved in dissection process for n-butane. First we erase the four vertices vertices, one at a time. This generated four subgraphs shown in the middle part of Fig. 1. The derived subgraphs are, in fact, the Ulam's subgraphs of n-butane. Ulam’s subgraphs arise in the famous, still unresolved, problem of the graph reconstruction.7, 8 In the next step we continue by dissection of the two propane fragments, which gives as fragments only isolated vertices and isolated edges signaling the end of step-wise dissection. By counting all isolated vertices (atoms) and isolated edges (bonds) we obtain the two parameters a and b, respectively. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 491 o—o—o—o o—o—o o o—o o—o o o—o—o o—o o o o—o o—o o o o—o Fig. 1 Dissection of n-butane: 6a + 6b Dissection of n-pentane isomers is illustrated in Fig. 2. After the first step we obtained as components n-butane, isobutane, and propane fragments. In the next step we dissected graphs of propane, butane, and isobutane, which as one can easily verify give: (2a + 2b), (6a + 6b) and (9a + 6b), respectively. With this information we obtain for the three pentane isomers of Fig.2 as the overall decomposition: (18a + 18b), (26a + 21b) and (40a +24b) respectively. In Table 1 we give the dissection parameters a, b for the five isomers of hexane, the nine isomers of heptane and the 18 isomers of octane. The a, b parameters for smaller alkanes were reported in ref. (5), but because of the numerical errors for 2,2-dimethylbutane, 2,3-dimethylbutane and 2-methylhexane the results for more branched heptanes and octanes were also in error so we give here the revised corrected values. The entries in Table 1 were ordered by increasing values of the parameters a, b. As we see M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 492 Acta Chim. Slov. 2000, 47, 489-506. the both parameters increase with molecular branching. Observe also that the range of the values assumed by a and b increases with the size of molecules. On this base alone one may expect few degeneracy, if any, which if shown to persist will make (a, b) parameters of interest in graph isomorphism problem. 9-11 + 2 + 2 (6a + 6b) + 2a + 2 (2a + 2b) + 2b 2 (6a + 6b) + 2a + b + a + (2a + 2b) + (9a + 6b) oo + 00 4 (9a + 6b) + 4a Fig. 2 Dissection of pentane, 2-methylbutane and 2,2-dimethylpropane 12-16 There are several graph invariants of a very high-resolution power -1 that may be of interest for graph isomorphism testing. The present invariants (a, b) are the first M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 493 integers that may show similar properties and as such deserve further attention in that respect -- which is not the subject of the present study. Table l: The dissection parameters a, b for the smaller alkanes Hexanes Octanes n-hexane 54 a + 54 b n-octane 486 a + 486 b 2-methyl 78 a + 68 b 2-methyl 702 a + 662 b 3-methyl 85 a + 74 b 3-methyl 850 a + 786 b 2,3-dimethyl 112 a + 88 b 4-methyl 905 a + 830 b 2,2-dimethyl 131 a + 94 b 2,5-dimethyl 1014 a + 914 b 3-ethyl 1072 a + 974 b Heptanes 2, 4-dimethyl 1241 a + 1096 b n-heptane 162 a + 162 b 2, 2-dimethyl 1313 a + 1099 b 2-methyl 234 a + 214 b 2, 3-dimethyl 1337 a + 1167 b 3-methyl 272 a + 245 b 3, 4-dimethyl 1520 a + 1314 b 3-ethyl 312 a +279 b 2-methyl-3-ethyl 1594 a + 1378 b 2, 4-dimethyl 338 a + 288 b 3, 3-dimethyl 1663 a + 1360 b 2, 3-dimethyl 395 a + 331 b 2, 2, 4-trimethyl 1917 a + 1550 b 2, 2-dimethyl 419 a + 324 b 3-methyl-3-ethyl 1948 a + 1581 b 3, 3-dimethyl 459 a + 359 b 2, 3, 4-trimethyl 1979 a + 1658 b 2, 2, 3-trimethyl 614 a + 460 b 2, 2, 3-trimethyl 2368 a + 1884 b 2, 3, 3-trimethyl 2535 a + 1945 b 2,2,3,3; tetramethyl 3708 a + 2772 b M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 494 Acta Chim. Slov. 2000, 47, 489-506. Use of the Dissection Parameters a, b for QSAR The importance of topological indices and other mathematical invariants of a chemical structural in QSAR rest solely on their use in structure-property and structure-activity studies. Topological indices may have additional use for detection of structural similarity / dissimilarity, , for characterization of molecular branching, -2 cyclicity, 30 'l'I 'I 4 35 36 37 molecular complexity, -3 graph density, the degree of convexity, molecular, , the degree of folding for a long chain structure,38-40 etc. Finally, they may be of interest in chemical documentation, including use in the isomorphism and the automorphism problems. The justification for extending study of mathematical properties of such indices, for which apparently some practitioners in the field do not give sufficient attention, follows from their use. If an index outperforms other indices, or combinations of other indices in a correlation of a particular property, it is of interest to investigate its property and in an effort to identify the critical structural elements that make the index successful. Hence, we have to postpone mathematical study of the dissection parameters till we can demonstrate its use in structure-property-activity studies. From Table 1 it is not immediately clear whether there is some regularity for the parameters a, b, except that they increase with molecular branching. It is also not apparent that such list of numbers has promise in structure-property studies. One way to find out is to test the new descriptors against a number of available molecular properties. However, it is possible in advance to assert whether these numbers, and for that matter any list of numbers associated with smaller alkanes, have promise in QSAR or not. All that is needed is to consult the "Periodic Table for Alkane Isomers." 41-43 In Table 2 we reproduce the "Periodic Table for Octane Isomers," which is derived from partial ordering of isomers based on the count of paths of length two and paths of length three, on which many physicochemical properties of alkanes depend. It was shown25, 41-43 that many properties as well as the values for many descriptors show a regular variation when their values for isomers are substituted in the corresponding site in the table. In Table 3 we show the variation of the parameter a obtained by dissection of octane isomers within the "Periodic Table." As we see the values increase from the top to the bottom and from the left to the right, with the only a minor numerical exception for M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 495 2, 3, 4-trimethylpentane and 3-methyl-3- ethylpentane. The values for 3 -methylhepthane and 4-methylheptane, which belong to the same cell of the periodic table, and the values for 2-methyl-3-ethylpentane and 3, 4-dimethylhexane which belong to another single cell, as we see from Table 3, differ little. This gives further support to the observed regular variation of the dissection parameters within the table. Table 2 Periodic table for octane isomers n-8 2-methyl 3-methyl 4-methyl 3-ethyl 2,5-dimethyl 2,4-dimethyl 2,3-dimethyl 3,4-dimethyl 2-methyl-3-ethyl 2,2-dimethyl 3,3-dimethyl 2,3,4-trimethyl 3-methyl-3-ethyl 2,2,4-trimethyl 2,2,3-trimethyl 2,3,3-trimethyl 2,2,3,3-tetramethyl Table 3 Periodic table for octane isomers with inscribed dissection parameter a 486 702 850 905 1072 1014 1241 1337 1520 1594 1313 1663 1979 1948 1917 2368 2535 3708 M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 496 Acta Chim. Slov. 2000, 47, 489-506. We tested the parameters (a, b) against several physicochemical properties of octanes. The results are summarized in Table 4 for properties associated with the correlation coefficient close to 0.9000 or higher. Table 4 Correlations of properties of octanes with descriptors (a, b) Property r (a, b) s (a, b) F (a, b) r (a) * see text Steric factor 0.9861 0.306 246 0.9486 R2 0.9164 0.077 39.3 0.9152* 0.904 Entropy 0.8982 1.60 29.2 0.8980 0.954 CP 0.8693 0.739 21.7 0.8636 0.668 In the last column we give the best regression coefficients as reported in ref [44] in which some 40 molecular descriptors were tested against some twenty physicochemical properties of octane isomers. We have employed two descriptors, a and b, hence the comparison may be somewhat biased. In order to make a more balanced comparison we included in the adjacent to the last column the regression coefficient when only the dissection parameter a is used. In the case of the quadratic mean radius we have used quadratic, rather than linear regression, (which yields low r value of 0.770). As we see from the short comparison only in the case of the entropy S better molecular descriptors than a, or a and b combined were reported in ref. [44]. The regression coefficient of around 0.900 is not necessarily satisfactory, though one should keep in mind that we are correlating properties for isomers which are molecules of the same size, thus making regression more difficult to fit, because the dominant size contribution has been effectively eliminated from considerations. Nevertheless, as we see from the above comparison the dissection parameters a alone, M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 497 and a, b jointly, produced much better regression for the quadratic mean radius R and the critical pressure (CP) than hitherto available alternative descriptors. The correlation power of new descriptors (a, b) has well been illustrated for the octane steric factors that have hitherto eluded simple correlation, or correlation based on few descriptors only. This definitely demonstrates use of (a, b) descriptors, which is further emphasized by the fact that the dissection parameters (a, b) are integers, the most simplest type of molecular descriptors, according to the classification of Balaban. 45 If we compare (a, b) with other integer molecular descriptors, the Wiener number, 46 the Hosoya's Z topological index, 47 and the hyper-Wiener index, 48, 49 we see that dissection parameters (a, b) are larger integers. This allows them to fit data with a greater precision that typifies the real valued topological indices, like the connectivity index 1%, 25 Balaban's J index, 50 and similar indices. Here we will illustrate correlation with Pitzer constants (P) for octanes. Linear correlation based on a as the only descriptor already gives satisfactory results for octane: P = - 0.00005086 a+ 0.411422 (1) with r = 0.9787, s = 0.0064, and F = 342. The correlation is shown in Fig. 3. The quadratic regression using the parameter a is marginally better than the linear regression. For the quadratic regression (shown in Fig. 4) we have: P = - 0.00008181 a + 0.00000001 a2+ 0.43151 (2) with r = 0.9858, s = 0.0054 and F = 243. The correlation of the parameter b against the parameter a for the 18 isomers of octane is shown in Fig. 5. The correlation between the two descriptors shows considerable duplication: r = 0.9981, s = 34.71, and F = 4228. It appears that one of the descriptors suffices for regression, while the other does not bring much novelty. One can combine the two descriptors as a combination (a + 2b) which represents a result obtained by further dissection of the edge as a fragment. However, the descriptor (a + 2b) does not improve the result over the allergy found regression using a as a single descriptor. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 498 Acta Chim. Slov. 2000, 47, 489-506. 400 380 -• 360 -• 340 -• 320 -• 300 -• 280 Fig. 3 Linear regression of Pitzer factor of octane against a P t z e r 400 380 360 340 320 300 280 Fig. 4 a x 10^3 Quadratic regression of Pitzer factor of octane against a M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 499 Use of the descriptor b alone gives results similar to those obtained with descriptor a: P = - 0.00010074 b + 0.00000001 b 2 + 0.440409 (3) with statistical parameters: r = 0.9787, s = 0.0061 and F = 159. Even this result is better than regression using both a and b as descriptors, which gives: P = - 0.00005218 a + 0.00000183 b + 0.411096 (4) with r = 0.9787, s = 0.0066 and F = 159. Fig. 5 The correlation of the parameter b against the parameter a for the 11 isomers of octane Regression for Steric Factors Recently introduced shape indices 2, 3 p2/w2, p3/w3, p4/w4 were successfully correlated with several molecular properties. Here we wish to consider in particular the regressions with the steric factors. The the statistical parameters when using two and three shape indices respectively are shown in Table 5. The results are comparable with M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 500 Acta Chim. Slov. 2000, 47, 489-506. those based on the dissection parameters (a, b), but even three shape indices do not give regression as good as that based on the dissection parameters. Table 5 Statistical parameters for path/walk descriptors Descriptors r s F p2/w2, p3/w3 0.9693 0.453 109 p2/w2, p3/w3, p4/w4 0.9710 0.457 71 400 -i-------------------------------------------------------------------------------------------------i 380 -• S^ 360 -• ¦ s^ ^r m m ^r 340 -- sS ^r m m jiT 320 -• yS 300 "• yS 280 4-----------1-----------1-----------1-----------1-----------1----------- 280 300 320 340 360 380 400 Fig. 6 The linear regression of the calculated values of the steric factor against the experimental values A quadratic relation of the hyper-Wiener index WW with the steric factors is quite satisfactory: r = 0.9821, s = 0.346 and F = 191. This compares well with the quadratic M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 501 regression based on either the parameter a alone, or b alone (r = 0.9715 and r = 0.9776 respectively) and approaches the regression based on (a, b). In Table 4 we report the experimental and the calculated values for steric factors of octanes as obtained in the regression using the parameters (a, b). The regression is shown in Fig. 6. Table 6 Experimental values for steric factors, the calculated values and the residuals of a regression using the dissection parameters (a, b). Molecule Steric (exp) Steric (calc) Residual n-octane 0 -0.297 0.297 2-methylheptane 0.608 0.489 0.119 3-methylhepfane 1.216 1.076 0.140 4-methylheptane 1.216 1.264 -0.048 3-ethylhexane 1.824 1.984 -0.160 2,2-dimethylhexane 1.397 1.851 -0.454 2,3-dimethylhexane 2.818 2.623 0.195 2,4-dimethylhexane 1.824 2.376 -0.552 2,5-dimethylhexane 1.216 1.593 -0.376 3,3-dimethylhexane 2.794 2.782 0.012 3,4-dimethylhexane 3.513 3.259 0.254 2-methyl, 3-ethylpentane 3.513 3.581 -0.068 3-methyl, 3-ethylpentane 4.191 3.660 0.531 2,2,3-trimethylpentane 4.493 4.632 -0.139 2,2,4-trimethylpentane 3.743 3.466 0.277 2,3,3-trimethylpentane * * * 2,3,4-trimethylpentane 4.598 4.504 0.094 2,2,3,3-tetramethylbutane 6.501 6.622 -0.121 M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 502 Acta Chim. Slov. 2000, 47, 489-506. The largest residuals are obtained for 3-methyl-3-ethylpentane, 2,4-dimethylhexane and 2,2-dimethylhexane. The experimental value of the steric factor for 2, 3, 3-trimethylpentane was not reported. The equation for the correlation of the calculated and the experimental values of the steric factor is very close to equation y = x (linear coefficient is 0.97233 and the constant is 0.07399). The coefficient of the correlation is 0.9861, the standard error 0.292 and the Fisher ratio F = 572. Correlation of (a, b) with other Topological Indices The dissection parameters (a, b) do not correlate with most of other topological indices. This could be expected in view that steric factor could not be well correlated with the existing topological indices. We do find however that (a, b) correlate quite well with WW index, the hyper-Wiener index. The parallelism between (a, b) and WW is confirmed by correlating WW with a, b, which gives the linear regression equation: WW = 0.5305 a - 0.879 b + 576.54 (5) and the statistical parameters: r = 0.9494, s = 26.5, and F = 69. If so computed WW is plotted against WW we have a quadratic regression equation: WW calc =1.9030 WW - 0.0018 WW2 -121.11 (6) and the statistical parameters : r = 0.9686, s = 19.9, and F = 114. Hence, we may expect that the dissection parameters (a, b) will be of interest in correlations in which hyper-Wiener index WW has been found suitable. The WW index was tested against doze physicochemical properties of octanes, 41 however, as a single descriptor it was not better than the Wiener number W, the JJ, or the leading eigenvalue of the Wiener matrix. Only in combination with other indices WW resulted in some fine regression. It seems therefore that the novel descriptors, the dissection parameters (a, b), could be of interest in regression in which they are similarly combined with other descriptors, in particular with the descriptors that combined with WW produced very good correlations, as shown in ref [49]. We hope to explore these novel possibilities but such work is outside the scope of the present report. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 503 Open Problems In conclusion we may say that one is justified to extend the present study and explore properties of novel molecular descriptors. We already have preliminary results for larger trees and smaller cyclic graphs. In addition we obtained the recursion relations for selected homologous series of molecules. The particularly simple expression holds for n-alkanes and 2-ethylalkanes. For the former, an = bn = 6 n, while for the latter we have separate recursion for an and bn: an+1 = 3 an bn+1 = 3 bn + 10 n For trees we have an > bn, except for chains when an = bn. In cyclic compounds, with or without the pending branches, we can have an > bn, an < bn, and an = bn. This last relation always holds for monocyclic Cn structures. Among the open problems the very important is the question of the uniqueness of (a, b) characterization of trees. If this can be proved there would be no need for exhaustive search for the smallest pair of trees with duplicate values for (a, b). While (a, b) characterization may be unique for trees, the situation may not hold for cyclic graphs. Table 7 The contribution to the dissection parameters (a, b) of the symmetry nonequivalent carbon atoms of different hexane isomers Isomer atom 1 atom 2 atom 3 atom 4 atom5 n-octane 18a + 18b 7a +6b 2a + 3b 2-methyl 18a + 18b 4a +2b 2a + 3b l0a + 6b 26a + 21b 3-methyl 26a + 21b 7a + 6b a + 2b 18a + 18b 2,3-dimethyl 26a + 21b 4a + 2b 2,2-dimethyl 26a + 21b 3a + b 10a + 6b 40a + 24b Another question that merits attention is the problem of local variations of (a, b). Because the parameters are additive in terms of Ulam's subgraphs one can partition the M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 504 Acta Chim. Slov. 2000, 47, 489-506. molecular (a, b) to individual atoms. In Table 7 we illustrate such partitioning for the individual (symmetry non-equivalent) atoms of five hexane isomers. As we see from these few illustrations the contributions of the terminal atoms is much greater than that of the internal atoms. There is here some parallelism with the connectivity index %, which assigns to terminal CC bonds greater weight then to the internal CC bonds. We can contrast this to the components entering the construction of the Wiener index W, and the hyper-Wiener index, where the internal bonds make a greater contribution. Acknowledgements This work was supported in part by the grant Jl-8901-0104-97 from the Ministry of Science and Technology of Slovenia. We thank the National Institute of Chemistry for supporting the visit of M. Randić to Ljubljana. We also express our thanks to NSFC (National Natural Science Foundation of China) and CSC (China Scholarship Council) for supporting this research and the visit of X. Guo to Drake University. References and Notes 1 M. Randić, On characterization of the shape of molecular graphs, J. Chem. Inf. Comput. Sci., (submitted) 2 M. Randić, Novel shape descriptors for molecular graphs (to be published) 3 M. Randić, The first graph theoretical descriptor of chirality, J. Chem. Inf. Comput. Sci., (submitted) 4 M. Randić, A missing descriptor p3* for QSAR, J. Chem. Inf. Comput. Sci., (submitted) 5 M. Randić, On dissection of acyclic graphs, MATCH, 1979, 5, 135-148. 6 M. Randić and W. L. Woodworth, Characterization of acyclic graphs by successive dissection, MATCH, 1982, 13, 291-313. 7 S. M. Ulam, A Collection of Mathematical Problems, Wiley, New York (1960). 8 J. A. Bondy and R. L. Hemminger, Graph reconstruction -- a survey, J. Graph Theory, 1977, 3, 227. 9 R. C. Read and D. G. Corneil, The graph isomorphism disease, J. Graph Theory, 1977, 1, 339-363. 10 G. Gati, Further annotated bibliography on the isomorphism disease, J. Graph Theory, 1979, 3, 95-109. 11 A. A. Oliferenko, V. A. Palyulin and N. S. Zefirov, The graph isomorphism disease: Twenty years after (Preprint) 12 M. Randić, On molecular identification numbers, J. Chem. Inf. Comput. Sci.,1984, 24, 164-175. 13 M. Randić, Molecular ID numbers: By design, J. Chem. Inf. Comput. Sci., 1986, 64, 134-136. 14 K. Szymanski, W. R. Müller, J. V. Knop and N. Trinajstić, Molecular ID numbers; Croat. Chem. Acta, 1986, 59, 719-723: 15 C.-Y. Hu and L. Xu, Developing molecular identification numbers by an all-path method, J. Chem. Inf. Comput. Sci., 1997, 37, 311-315. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes Acta Chim. Slov. 2000, 47, 489-506. 505 16 C.-Y. Hu and L. Xu, On highly discriminating topological index, J. Chem. Inf. Comput. Sci.,1996, 36, 82-90. 17 M. Randić, Design of molecules with desired properties. Molecular similarity approach to property optimization, in: Concepts and Applications of Molecular Similarity, (M. A. Johnson and G. Maggiora, Eds.), Wiley, New York (1990), pp. 77-145. 18 M. Randić, Similarity methods of interest in chemistry, in: Mathematical Methods in Contemporary Chemistry (I. S. Kuchanov, Ed.), Gordon and Breach Publ. Inc, (1995), pp.1 -100. 19 M. Randić, On characterization of molecular branching, J. Am. Chem. Soc, 1975, 97, 6609-6615. 20 D. Bonchev and N. Trinajstić, Information theory, distance matrix and molecular branching, J. Chem. Phys.,1977, 67, 4517 5433. 21 D. Bonchev and N. Trinajstić, On topological characterization of molecular branching, Int. J. Quant. Chem: Quant. Chem. Symp.,1978, 12, 293-303. 22 S. H. Bertz, Branching in graphs and molecules, Discrete Appl. Math., 1988,19, 65-83. 23 D. H. Rouvray, The challenge of characterizing branching in molecular species, Disc. Appl. 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Randić and G. Krilov, On characterization of folding of proteins, Int. J. Quant. Chem. 1999, 75, 1017-1026 41 M. Randić and C. L. Wilkins, Graph theoretical basis for ordering of structures, Chem. Phys. Lett, 1979, 63, 332-336. 42 M. Randić and C. L. Wilkins, Graph theoretical ordering of structures as a basis for systematic searches for regularities in molecular data, J. Phys. Chem., 1979, 83, 1525-1540. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes 506 Acta Chim. Slov. 2000, 47, 489-506. 43 M. Randić, Chemical structure - What is she? J. Chem. Educ.,1992, 69, 713-718. 44 M. Randić, Comparative regression analysis. Regressions based on a single descriptor, Croat. Chem. Acta,1993, 66, 289-312. 45 A. T. Balaban, Lowering the intra- and intermolecular degeneracy of topological invariants, Croat. Chem. Acta,1993, 66, 447-458. 46 H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc, 1947, 69, 17-20. 47 H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Japan, 1971, 44, 2332-2339. 48 M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett, 1993, 221, 478-483. 49 M. Randić, X. Guo, T. Oxley, H. Krishnapriyan and L. Naylor, Wiener matrix invariants, J. Chem. Inf. Comput. Sci., 1994, 34, 361-367. 50 A. T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett., 1982, 89, 399-404. Povzetek Disekcija grafov, ki je vpeljana pred dvajsetimi leti, daje karakterizacijo molekularnih grafov s celimi števili. Ta pristop, ki vnaša nekatera matematična vprašanja, omogoča uvajanje novih strukturnih deskriptorjev, ki so potencialno zanimivi za QSAR študije. V tem delu je disekcija grafov ponovno raziskana in nekaj napak zgodnejših objav je popravljenih. Novi deskriptorji, dobljeni s procesom disekcije, so ilustrirani na izbranih lastnostih oktanov. M. Randić, X. Guo, P. Calkins: Graph dissection revisited. Application to smaller alkanes