726 Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... DOI: 10.17344/acsi.2019.5346 Scientific paper Usefulness of Combinations of Vertex-Degree Weighted Path Indices and Elements of a Universal Matrix Anton Perdih Faculty of Chemistry and Chemical Technology, University of Ljubljana (retired) Večna pot 113, 1000 Ljubljana, Slovenia * Corresponding author: E-mail: a.perdih@gmail.com Received: 06-19-2019 Abstract The mutually optimized combinations of vertex-degree weighted path indices and the vertex-degree vertex-distance weighted elements of the Universal matrix were applied in the way of TInew = ∑kN×PN(aN,bN,...) + kij×uij(aij,bij,cij). They were correlated to the boiling points of octanes. Using the mutually optimized combination of vertex-degree weighted path indices P2(a2,b2,c2), P3(a3,b3,c3,d3), and P4(a4,b4,c4,d4,e4) there was observed R = 0.99950. When in addition to P2(a2,b2,c2), P3(a3,b3,c3,d3), and P4(a4,b4,c4,d4,e4) also P1(a1,b1) or u72(a72,b72,c72) or u76(a76,b76,c76) or u62(a62,b62,c62) or u74(a74,b74,c74) or u52(a52,b52,c52) have been applied, then the goodnesses of up to R = 0.99988, S = 0.098, IC = 98.5% have been observed. The mutually optimized combinations of vertex-degree weighted path indices and the vertex-degree vertex-dis- tance weighted elements of the Universal matrix are promising indices also for other physicochemical properties of octanes. Keywords: Boiling point; Octanes; Matrix elements; Path indices; Structural interpretation 1. Introduction Path indices1,2 were first used by Wiener3,4 and to- gether with the connectivity indices5-8 they are being used by numerous other authors. The path indices1,2 and the connectivity indices5-8 were previously considered to be two different groups of indices. It has been, however, shown9 that both groups of them are parts of the ver- tex-degree weighted path indices PN(a, b, ...). The mutu- ally optimized combinations of vertex-degree weighted path indices gave rise in the case of 29 physicochemical properties of octanes to R (P1..P4) > 0.9.9 In the mutu- ally optimized combinations with some of the elements of the Universal matrix the values were R = 0.994 or higher. Ivanciuc10,11 described the Dval matrix and present- ed its characteristics. His approach was developed into the Universal matrix by using the vertex-degree vertex-dis- tance weighted elements. The indices derived from it were tested for their usefulness as descriptors of a number of physicochemical properties of alkanes in general and of octanes in particular.12,13 For the definition of the Univer- sal matrix and its elements see Appendix 3 in ref.13 The mutually optimized combinations of the vertex-degree vertex-distance weighted elements of the Universal matrix were tested for their usefulness as descriptors of a number of physicochemical properties of octanes.14 Here is presented the approach using the mutually optimized combinations of vertex-degree weighted path indices and the vertex-degree vertex-distance weighted el- ements of the Universal matrix. The correlations to the boiling points (BP) of octanes are used to illustrate the usefulness of the approach, which is generally applicable. These combinations allow to be made also some structural interpretations of indices involved in them. 2. Definitions, Data and Methods Vertex-degree weighted path indices are defined as PN(aN, bN, ...) = ∑via×vjb×..., where vi, vj, etc. are the vertex degrees.9 The vertex-degree vertex-distance weighted ele- ments of the Universal matrix are defined as uij(aij, bij, cij) = via×vjb×dijc, where vi and vj are the vertex degrees and dijc is the distance between the vertices i and j.14 In the text, they are often presented in a shorthand form as PN and uij, respectively. 727Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... The mutual optimization of these indices and matrix elements is performed in the way of TInew = ∑kN×PN(aN, bN,..) + kij×uij(aij, bij, cij) using one or more of the ver- tex-degree weighted path indices PN(aN, bN,...) and/or the vertex-degree vertex-distance weighted elements of the Universal matrix uij(aij, bij, cij), where ∑|kN| + |kij| = 1. The indication of mutual optimization is presented in the text or tables mainly in a shorthand form. For example, the form P2 & P3 & P4 & u72 is indicating the optimization of TInew = k2×P2(a2, b2, c2) + k3×P3(a3, b3, c3, d3) + k4×P4(a4, b4, c4, d4, e4) + k72×u72(a72, b72, c72). The values of exponents a, b, ... and the value of the smallest of the factors k are taken here generally to have two significant digits. When a different number of significant digits is employed, this is explicitly mentioned. When nec- essary, one decimal more is used than the number of signif- icant digits, for example 0.092 in the case of two significant digits or 0.0921 in the case of three significant digits. As measures for the goodness of the linear correla- tion the following are used: the correlation coefficient R, the standard error S and the information content regard- ing the influence of branching14 IC. In most cases only the correlation coefficient R is given. The software for statistics calculations included in the program package MS Excel was used. The values of boiling points of octanes were taken from Li15 converting °C to K, ref.14 Appendix 1. 3. Results and Discussion The aim of this paper is to demonstrate in the case of the boiling points of octanes the way of how to proceed to develop the best possible descriptors using the vertex-de- gree weighted path indices in combination with the ver- tex-degree vertex-distance weighted elements of the Uni- versal matrix. In the first step, the values of exponents in individual vertex-degree weighted path indices as well as vertex-de- gree vertex-distance weighted elements of the Universal matrix are optimized. For the individual vertex-degree weighted path indices this is demonstrated in Table 1. The same should be done for the individual vertex-degree ver- tex-distance weighted elements of the Universal matrix. The best-observed values of the correlation coefficient R between optimized individual vertex-degree vertex-dis- tance weighted elements of the Universal matrix and the boiling point of octanes using two significant digits in their exponents are –0.82 < R < –0.88, thus between those of P2 and P4 in Table 1. In the second step, the mutually optimized combina- tions of two of them should be made. The best results for vertex-degree weighted path indices are presented in Table 2. The best individual result gives the vertex-degree weighted path three index P3(–0.9739, –1.3213, –1.667, –0.1944) with R < 0.95 as presented in Table 1. In the mu- tually optimized combination of two of vertex-degree weighted path indices (Table 2) the best result is observed using the vertex-degree weighted path indices P2 & P4 in the form TInew = –0.519×P2(–0.421, –0.408, –0.304) –0.481×P4(0.364, –0.075, –0.728, –0.705, –∞) giving rise to R > 0.99. The procedure is continued including additional ver- tex-degree weighted path indices or vertex-degree ver- tex-distance weighted elements of the Universal matrix. The mutually optimized combination of four ver- tex-degree weighted path indices using two significant digit values in exponents and the smallest factor k in the form of TInew = P1 & P2 & P3 & P4 ; k1*P1(a1, b1) + k2*P2(a2, b2, c2) + k3*P3(a3, b3, c3, d3) + k4*P4(a4, b4, c4, d4, e4) is TInew = –0.0033×P1(–0.40, 1.88) –0.6537×P2(–0.087, –0.68, –0.007) –0.2947×P3(0.004, –0.56, –1.89, –7.3) –0.0483×P4(0.45, –0.005, –0.52, 0.28, 0.91), giving rise to R = 0.999579, S = 0.183, IC = 97.1%. Using three significant digits, it is TInew = –0.0257×P1(–0.1799, 1.595) –0.6522×P2(–0.0266, –0.680, –0.0233) –0.2759×P3(0.0266, –0.618, –1.876, –∞) –0.0462×P4(0.457, –0.0966, –0.535, 0.278, 1.076) giving rise to R = 0.999763, S = 0.137, IC = 97.8%. This demon- strates that it is advisable to use at least two or, even better, three significant digits. Table 1. The values of the correlation coefficient R between opti- mized individual vertex-degree weighted path indices and the boil- ing point of octanes as a function of the number of significant digits in their exponents. In bold: the best of them. Significant R(PN, BP) digits P1 P2 P3 P4 grid14 –0.8961 –0.9083 0.9127 0.7967 one –0.8961 –0.9095 0.9424 0.7970 two –0.9125 –0.9107 0.9433 0.7973 three –0.9161 –0.9107 0.9433 0.7974 four –0.9175 –0.9107 0.9433 0.7974 Table 2. The goodness of correlation between mutually optimized combinations of the vertex-degree weighted path indices and the boiling point of octanes. TInew R S IC (%) Sign. dig. P2 & P3 0.9797 1.266 79.9 2 0.9804 1.241 80.3 3 P2 & P4 0.9926 0.767 87.8 2 0.9928 0.755 88.0 3 P3 & P4 0.9803 1.246 80.3 2 0.9870 1.013 83.9 3 P2 & P3 & P4 0.9956 0.588 90.7 2 0.99950 0.202 96.8 3 P1 & P2 & P3 & P4 0.999579 0.183 97.1 2 0.999763 0.137 97.8 3 Sign. dig. - No. of significant digits in exponents a, b, ... and the smallest factor k In bold - The best of that group 728 Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... The mutually optimized combinations of a ver- tex-degree weighted path index with particular vertex-de- gree vertex-distance weighted elements of the Universal matrix are presented in Figure 1 for P2(a2, b2, c2). For the vertex-degree weighted path indices P1(a1, b1), P3(a3, b3, c3, d3), and P4(a4, b4, c4, d4, e4) they are presented in Ap- pendix 2. Figure 1. Correlation coefficients of the mutually optimized combi- nations of the vertex-degree weighted path two index with ver- tex-degree vertex-distance weighted elements of the Universal ma- trix uij(aij, bij, cij). As can be seen in Figure 1 and in additional Figures in Appendix 2, some of the vertex-degree vertex-distance weighted elements of the Universal matrix improve sub- stantially the goodness of correlation obtained by the indi- vidual vertex-degree weighted path indices. In combinations of one vertex-degree weighted path index with one vertex-degree vertex-distance weighted el- ement of the Universal matrix the most promising are on average the matrix elements u63(a63, b63, c63), u75(a75, b75, c75), u32(a32, b32, c32), u71(a71, b71, c71), u72(a72, b72, c72), and u53(a53, b53, c53). Table 3 demonstrates that some of the vertex-degree vertex-distance weighted elements of the Universal matrix improve the optimized combination of the vertex-degree weighted path indices P2 & P3 & P4 better than the ver- tex-degree weighted path one index. The best of them is TInew = P2 & P3 & P4 & u72 ; k2*P2(a2, b2, c2) + k3*P3(a3, b3, c3, d3) + k4*P4(a4, b4, c4, d4, e4) + k72*u72(a72, b72, c72) = –0.69059*P2(–0.0132, –0.675, 0.0435) –0.04476*P4(0.404, 0.0793, –0.498, 0.288, 1.106) Table 3. The best observed results in the mutually optimized com- binations of vertex-degree weighted path indices with the ver- tex-degree vertex-distance weighted elements of the Universal ma- trix in relation to the boiling points of octanes using three significant digits. R S IC (%) P2 & P3 & P4 & u72 0.99988 0.098 98.5 P2 & P3 & P4 & u76 0.99983 0.115 98.2 P2 & P3 & P4 & u62 0.99983 0.117 98.1 P2 & P3 & P4 & u74 0.99981 0.123 98.0 P2 & P3 & P4 & u52 0.99978 0.133 97.9 P2 & P3 & P4 & P1 0.99976 0.137 97.8 P2 & P3 & P4 0.99950 0.199 96.8 –0.25706*P3(–0.1118, –0.601, –1.905, –2.53) –0.00759*u72(–2.08, 1.137, 0.0315) giving rise to R = 0.99988, S = 0.098, IC = 98.5%. Its values are presented in Table 4. In the case of octane isomers its values are not degenerated. Table 4. Values of the descriptor TInew = P2 & P3 & P4 & u72 round- ed to five decimals. But one should not forget that the values of the de- scriptor TInew = P2 & P3 & P4 & u72 are good in relation to the boiling points of octanes. For other properties of oc- tanes, some different mutually optimized combinations of vertex-degree weighted path indices with the vertex-de- gree vertex-distance weighted elements of the Universal matrix are to be developed in the way demonstrated above. One of the earliest descriptor combinations used was the combination of the Wiener index (W) with the path three index (p3).3,4 Correlations in combinations of de- scriptors p3 and W are presented in Table 5. Octane TInew Oct –3.09185 2M7 –3.19772 3M7 –3.18390 4M7 –3.20116 3Et6 –3.18612 25M6 –3.31415 24M6 –3.31140 23M6 –3.22836 34M6 –3.19898 3Et2M5 –3.22757 22M6 –3.34633 33M6 –3.27560 3Et3M5 –3.19138 234M5 –3.25649 224M5 –3.44826 223M5 –3.30526 233M5 –3.23618 2233M4 –3.35245 729Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... Table 5. Correlations of the combinations of descriptors p3 & W with the boiling points of octanes (TInew = k3*p3 + kw*W; k3 + kw = 1). R S IC (%) k3 = kW = 0.5 0.6740 4.660 26.1 optimized k3 * 0.9888 0.943 85.1 optimized k3 ** 0.9892 0.926 85.3 * two significant digits ** three significant digits They are not as good as in the mutually optimized combination of P2 & P4 & P3 & u72. In comparison of optimized combinations p3 & W vs. P2 & P4 & P3 & u72 it is R = 0.9892 vs. 0.99988, S = 0.926 vs. 0.098, and IC (%) = 85.3 vs. 98.5. The results of the optimized combinations of ver- tex-degree weighted path indices with the indices W,7,8 RW,16 and χ,5 are presented in Table 6. For comparison, the goodness of the correlation of the optimized combination of indices W,7,8 RW,16 and χ,5 is R = 0.9565, S = 1.840. Table 6. Best correlations with boiling points of octanes of the com- binations of indices W, RW, and χ with the optimized vertex-degree weighted path indices, R(BP, PN), using two significant digits in factors k and exponents. In bold: R > 0.95. PN &: W RW χ P1 0.9421 0.9175 0.9142 P2 0.9124 0.9174 0.9295 P3 0.9892 0.9696 0.9447 P4 0.7990 0.7976 0.8609 Best correlation with BP of the optimized combina- tion of index W with the vertex-degree weighted path three index P3(a3,b3,c3,d3) taking three significant digits in factors k and exponents, is: R = 0.9931, S = 0.738, IC = 88.3 (%). The goodness of correlations presented here in Table 3 is, regarding S, between one and two orders of magni- tude better than those presented in Tables 4–7. Table 7. Goodness of correlation of descriptor combinations with the boiling point data of octanes Author(s) Ref. No. of indices R S Ivanciuc et al.17 3 0.994 2.79 Ivanciuc et al.18 2 0.984 4.94 Randić19 2 0.914 2.56 This work 4 0.99988 0.098 The goodness of correlation of individual path/walk indices19 with boiling point data of octanes is low, |R| < 0.71, S > 4.5. The goodness of R > 0.9 and S < 2.6 gives the optimized combination of p2/w2 & p3/w3,19 Table 7. Com- paring it to p2 & p3 (R = 0.919, S = 2.49) it is a slightly less good descriptor for the boing point of octanes, and com- paring it to P2 & P3, Table 3, (R = 0.9797, S = 1.266) it is much less good. Better than p2/w2 & p3/w3,19 are the optimized com- binations p2/w2 & p3/w3 & p5/w5 (R = 0.9206, S = 2.463) and p2/w2 & p3/w3 & p4/w4 (R = 0.9780, S = 1.315). Com- paring the latter to p2 & p3 & p4 (R = 0.9674, S = 1.598) indicates that the combinations of the path/walk indices are more promising than the combinations of the path in- dices. Comparing to P2 & P3 & P4, Table 3, (R = 0.99950, S = 0.202), the optimized combination of p2/w2 & p3/w3 & p4/w4 is much less good than the corresponding optimized combination of the vertex-degree weighted path indices. The optimized combination of four path/walk indi- ces of octanes, TInew = ∑ki×pi/wi = k2*p2/w2 + k3*p3/w3 + k4*p4/w4 + k5*p5/w5 = –0.403*p2/w2 + 0.115*p3/w3 –0.305*p4/w4 –0.177*p5/w5 gives rise to R = 0.9883, S = 0.962, which is better than the optimized combination of p2 & p3 & p4 & p5 (R = 0.9780, S = 1.317) and slightly better than the combination P3 & P4, Table 2. Among the vertex-degree vertex-distance weighted elements of the Universal matrix, which improve the cor- relation of P2 & P3 & P4 more than P1, Table 3, there are u72 > u76 > u74 and u72 > u62 > u52. In the five vertex-degree vertex-distance weighted elements of the Universal ma- trix, which improve the correlation of P2 & P3 & P4 better than P1, there appear the vertices No. 2 and No. 7 three times each, vertex No. 6 two times, and vertices No. 4 and 5 once. Thus, the vertices No. 2 and 7 contribute the larger part of the improvement, whereas the vertices No. 4, 5, and 6 contribute less. The structural interpretation of the contribution of vertices No. 2, 4, 5, 6, and 7 can be based on two criteria, i.e. whether they are interior or peripheral vertices. The in- terior vertices are either bearing the branches and they are thus contributing the information about the branching of octanes or they are exposed to intermolecular contacts, which are modified by the sterical hindrance of neigh- bouring branches. The peripheral vertices are exposed to and modifying the intermolecular contacts. For illustration is presented the structure of 2,3-di- methylhexane: 2,3-dimethylhexane: In 2,3-dimethylhexane, the vertices No. 1, 6, 7, and 8 are of degree one, the vertices No. 4, and 5 are of degree two, the vertices No. 2 and 3 are of degree three, whereas there is no vertex of degree four. Among octane isomers, the vertices No. 1 and 8 are in all cases peripheral. The vertex No. 2 is in all cases an interior vetrex. By the IUPAC Chemical Nomenclature 730 Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... used here for the enumeration of vertices, the vertex No. 2 is of degree two in 7 out of 18 cases, of degree three in 7 out of 18 cases as well, and of degree four in 4 out of 18 cases. The same holds true for the vertex No. 3. They are bearing the majority of the information about branching of oc- tanes. However, the vertex-degree vertex-distance weight- ed elements of the Universal matrix containing the vertex No. 3 give rise to R between 0.9995 and 0.9996 and S be- tween 0.199 and 0.177, whereas the vertex-degree ver- tex-distance weighted elements of the Universal matrix containing the vertex No. 2 mentioned in Table 3 give rise to R between 0.99978 and 0.99988 and S between 0.133 and 0.098. The vertex No. 4 is peripheral in one case only, when it is of degree one. In other cases, it is an interior vertex. It is of degree two in 12 out of 18 cases, of degree three in 5 out of 18 cases, and in no case it is of degree four. The vertex No. 5 is interior in 11 out of 18 cases, whereas it is peripheral in 7 out of 18 cases. It is of degree three in one case only. The vertex No. 6 is peripheral in 14 out of 18 cases and when interior, it is of degree two only. The vertex No. 7 is of degree one among all octanes with the exception of n-octane, thus in 17 out of 18 cases. It is thus mainly a peripheral vertex exposed to intermo- lecular contacts and modifying them. The boiling point is dependent on intermolecular at- traction. Among octanes, it is dependent on attraction and repulsion between the structural groups C, CH, CH2, and CH3, i.e. speaking in terms of mathematical chemistry, be- tween vertices of degree four, three, two and one. Whereas at the equilibrium distance the contribution to attraction between equal functional groups is C >> CH > CH2 > CH3, its dependence on intermolecular distance d is d–6.20 The interplay of different contribution to attraction by struc- tural groups, which are represented in topological indices by vertices of different degrees, and of different distances between them in different liquid octane isomers, contrib- utes to differences in boiling points of octane isomers. The structural interpretation of the contribution of different vertices reflects this interplay. The differences between the experimental data of the boiling points of octanes and the calculated ones using combinations presented in Table 3 are shown in Figure 2. In Figure 2 we see that in the optimized combination of vertex-degree weighted path indices P2 & P3 & P4 the difference between the experimental data and the calculat- ed ones is larger than 0.1 K at octane isomers n-octane, 2-methylheptane, 3-methylheptane, 2,4-dimethylhexane, 2,3-dimethylhexane, 3-ethyl-2-methylpentane, as well as among all octanes having three resp. four branches. In the best observed case, the optimized combination of P2 & P3 & P4 & u72, the difference between the experimental data and the calculated ones is larger than 0.1 K at octane iso- mers 2-methylheptane, 4-methylheptane, 3-ethylhexane, 2,3,3-trimethylpentane, and 2,2,3,3-tetramethylbutane. If we compare the results observed here in the mutu- ally optimized combination of P2 & P3 & P4 & u72 with those reported by some other authors using more than one variable in their equations,17–19 we can see that the results obtained here are better than those by using other models with several (2 to 3) other indices at once. As a conclusion it can be said that the mutually opti- mized combinations of the vertex-degree weighted path indices and the vertex-degree vertex-distance weighted elements of the Universal matrix in the form of TInew = ∑kN×PN(aN, bN,...) + kij×uij(aij, bij, cij) give rise in the case of the boiling points of octanes up to R = 0.99988, S = 0.098, IC = 98.5% and similar combinations are thus promising indices also of other properties. That the ap- proach demonstrated here in detail is generally applicable has been shown previously for the mutually optimized combinations of the vertex-degree vertex-distance weight- ed elements of the Universal matrix14 as well as for the mu- tually optimized combinations of vertex-degree weighted path indices.9 There can be seen that at different physico- chemical properties of octanes different mutually opti- mized combinations of the vertex-degree weighted path indices,9 respectively the vertex-degree vertex-distance weighted elements of the Universal matrix14 give rise to the best correlation. 4. References 1. J. R. Platt, J. Chem. Phys. 1947, 15, 419–420. DOI:10.1063/1.1746554 2. M. Randić, J. Chem. Educ. 1992, 69, 713–718. DOI:10.1021/ed069p713 3. H. Wiener, J. Am. Chem. Soc. 1947, 69, 17–20. DOI:10.1021/ja01193a005 4. H. Wiener, J. Am. Chem. Soc. 1947, 69, 2636–2638. Figure 2. Differences between the experimental data of the boiling points of octanes and the calculated ones using correlations with P2 & P3 & P4 & uij. 731Acta Chim. Slov. 2019, 66, 726–731 Perdih: Usefulness of Combinations of Vertex-Degree Weighted ... DOI:10.1021/ja01203a022 5. M. Randić, J. Am. Chem. Soc. 1975, 97, 6609–6615. DOI:10.1021/ja00856a001 6. M. Randić, C.L. Wilkins, J. Phys. Chem. 1979, 83, 1525–1540. DOI:10.1021/j100474a032 7. L. B. Kier, L. H. Hall, J. Pharm. Sci. 1976, 65, 1806–1809. DOI:10.1002/jps.2600651228 8. L. B. Kier, L. H. Hall, Molecular Connectivity in Structure–Ac- tivity Analysis, Willey, New York, 1986. 9. A. Perdih, Acta Chim. Slov. 2016, 63, 88–96. DOI:10.17344/acsi.2015.1975 10. O. Ivanciuc, Rev. Roum. Chim. 1999, 44, 519-528. DOI:10.1097/00006123-199903000-00054 11. O. Ivanciuc, Rev. Roum. Chim. 2000, 45, 587-596. 12. A. Perdih, B. Perdih, Acta Chim. Slov. 2004, 51, 598-609. 13. A. Perdih, F. Perdih, Acta Chim. Slov. 2006, 53, 180-190. 14. A. Perdih, Acta Chim. Slov. 2015, 62, 879–888. DOI:10.17344/acsi.2015.1607 15. X. H. Li, Chem. Phys. Lett. 2002, 365, 135–139. DOI:10.1007/978-3-322-99997-9_9 16. M.V. Diudea, J. Chem. Inf. Comput. Sci. 1997, 37, 292–299. DOI:10.1021/ci960037w 17. O. Ivanciuc, T. Ivanciuc, A.T. Balaban, Internet Electron. J. Mol. Des. 2002, 1, 467–487. 18. O. Ivanciuc, T. Ivanciuc, D. Cabrol-Bass, A.T. Balaban, Inter- net Electron. J. Mol. Des. 2002, 1, 319–331. 19. M. Randić, J. Chem. Inf. Comput. Sci. 2001, 41, 607–613 DOI:10.1021/ci0001031 20. F. M. Fowkes, in R. L. Patrick, Ed., Treatise on Adhesion and Adhesives. Vol. 1: Theory, M. Dekker, New York, 1967, pp. 325–449. Povzetek Uporabljene so bile kombinacije uteženih indeksov poti in uteženih elementov Univerzalne matrike, optimirane na način TInovi = ∑kN×PN(aN,bN,...) + kij×uij(aij,bij,cij). Korelirane so bile z vrelišči oktanov. Kombinacije uteženih indeksov po- ti P2(a2,b2,c2), P3(a3,b3,c3,d3) in P4(a4,b4,c4,d4,e4) so dale korelacijo R = 0.99950. Ko so bili poleg uteženih indeksov poti P2(a2,b2,c2), P3(a3,b3,c3,d3) in P4(a4,b4,c4,d4,e4) uporabljeni tudi P1(a1,b1) ali u72(a72,b72,c72) ali u76(a76,b76,c76) ali u62(a62,b62,c62) ali u74(a74,b74,c74) ali u52(a52,b52,c52), so dali R do 0.99988, S do 0.098, IC do 98.5%. Optimirane kombi- nacije uteženih indeksov poti in uteženih elementov Univerzalne matrike so obetajoči indeksi tudi za druge fizikokem- ijske lastnosti oktanov. Except when otherwise noted, articles in this journal are published under the terms and conditions of the  Creative Commons Attribution 4.0 International License