180 Acta Chim. Slov. 2006, 53, 180–190 Scientific Paper Topological Indices Derived From Parts of a Universal Matrix Anton Perdih*, Franc Perdih Department of Chemistry and Chemical Technology, University of Ljubljana, Aškerčeva 5, SI-1000 Ljubljana, Slovenia. E-mail: Anton.Perdih@email.si Received 28-04-2006 Abstract Using the approach of mutual contribution of vertices to the value of topological indices derived from parts of a universal matrix gave the idea how to develop new topological indices. Some of the newly developed indices are useful indices of Octane Numbers and of Critical Pressure of alkanes of equal carbon number, giving rise to abs(Rfit) > 0.99. Using them we estimated the missing data of Octane Numbers of 2,3-dimethyl butane and 2,2,3,3-tetramethyl butane, as well as of nonanes and decanes. The best analogous indices derived from the reverse variants of this universal matrix are in general less useful. Some types of newly developed indices are also useful for structural interpretation of several known indices. Key words: alkane, index, octane number, structural interpretation, topological matrix Introduction In a previous study,1 several groups of indices were presented, which were derived from a generalized form of the vertex-degree vertex-distance matrix, denoted there as the G(a,b,c) matrix. This matrix is a universal matrix since it includes the distance matrix, the adjacency matrix, the reciprocal and several other types of matrices1-10 and one of the reviewers suggested that it should be better denoted as the U(a,b,c) matrix. From this matrix can be derived a number of previously known indices, for example the Wiener index11, the Randić index12 as well as those introduced in ref.13-22 In the same direction, by varying the exponent on the degree of vertices acted subsequently also Miličević and Nikolić,23 who developed the variable Zagreb indices. Their vM1 resp. vM2 indices are equal to some W(m,n) indices2 resp. to some Vij(m,n) indices.8 However, this universal matrix was introduced first by Ivanciuc13,24 as the Dval matrix and its properties were described as well.25 Therefore, wherever in present paper is mentioned the universal matrix, it is the Dval13,24 matrix, but the indices derived from it are presented in the I(a,b,c) notation (a ? q, b ? r, c ? p) for easier comparison with the previously1 tested ones. In present paper are considered topological indices derived from this matrix by summation of some of its non-diagonal elements. Few groups of them, i.e. the VC(a,b,c) and VNL(a,b,c) indices,1 as well as the ?V(a,b,c) indices1,7 were presented earlier. Here are introduced some additional groups of them. Data and notations Physicochemical properties (PP) The data for the boiling point (BP), density (d), the critical data Tc, Pc, Vc, Zc, ac, and dc, the standard enthalpy of formation for the ideal gas (AHf°g), the enthalpy of vaporisation (AHv), the Antoine constants A, B, and C, as well as the Pitzer's acentric factor (ro) and the refractive index (nD) were taken from the CRC Handbook26 or from Lange’s Handbook.27 The data for the liquid molar volume (Vm), the intrinsic molar volume (Vi), the intermolecular volume (V”), the ratios Tc2/Pc and Tc/Pc used instead of the van der Waals parameters a0 and b0, the ratio BP/Tc (reduced BP), the molar refraction (MR), cohesive energy density (CED) and its square root, the solubility parameter (Sol. par.) were calculated from data presented in the handbooks. The data for Octane Numbers (BON, MON, RON) were taken from: Balaban and Motoc,28 Gutman et al.,22 and Morley;29 those for vapour pressure (logVP) from Goll and Jurs,30 and those for the entropy (S) and quadratic mean radius (R2) from Ren.31 Surface tension (ST) data were taken from Li.32 Quality of results The quality of linear relationships is expressed by fitted statistical parameters: number of data in the set, N, the correlation coefficient, Rfit, the standard error of estimate, Sfit, and Fisher’s ratio, F. The results are also cross(internally)-validated by a leave-out-one, root mean square procedure, which is reflected in the sign of the cross-validated statistical parameters. Statistical Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix Acta Chim. Slov. 2006, 53, 180–190 181 parameters for the cross-validated linear relationships are denoted Rcv and Scv, where the subscript cv denotes cross-validation. Notations The structures of alkanes are presented in shorthand, e.g. Hp is n-heptane, Oct is n-octane, 223M5 is 2,2,3-trimethylpentane, 3E2M5 is 3-ethyl-2-methylpentane, etc. Derivation of additional groups of indices from parts of a universal matrix Lučić et al.33 demonstrated the usefulness of structural interpretation34-36 of topological indices for the development of new and better indices. However, structural interpretation introduced by Randić et al.34-36 should be performed with caution, since it may happen that incomparable results are compared.37 For the indices derived from some of the non-diagonal elements of the universal matrix, for example the VC(a,b,c), VNL(a,b,c), and ?V(a,b,c) indices,1 the principle of the mutual contribution of graph vertices to the value of index37 has been found appropriate to interpret these indices. Among these indices, the approach using the mutual contribution of graph vertices to the value of index gives rise to some symmetry related patterns in their structures, which are demonstrated in Figure 1 for pentanes. This type of symmetry relations is well expressed also among the structures of higher alkanes. The symmetric pattern observed in Figure 1 at 2M4 is observed generally at 2-methyl alkanes, 3-ethyl alkanes, etc. Another interesting case is 3-ethyl-2-methyl pentane, which gives the pattern of what would be 3-isopropyl pentane. Furthermore, vertices of degree one contribute mutually only to the VC(a,b,c) indices, together with several other cases of mutual contribution of vertices of equal degree. Some exclusive mutual contributions of vertex pairs of different degree to the VNL(a,b,c) and ?V(a,b,c) indices have been observed as well. Another reviewer raised the question, in which relation are mentioned indices to the symmetric and anti-symmetric part of the universal matrix. The answer is that the V(a,b,c) indices1,8 can be derived from the symmetric part of the universal matrix but they contain all the information present in the universal matrix. The ?V(a,b,c) indices,1,7 on the other hand, can be derived from the anti-symmetric part of the universal matrix and they do contain the information of the anti-symmetric part of it. The VC(a,b,c) or VNL(a,b,c) indices can not be derived from the symmetric or the anti-symmetric part of the universal matrix. They are defined to be derived from the left half of it.1 From the observation of the exclusive mutual contribution of vertex pairs of equal resp. different VC(a,b,c) VNL(a,b,c) n-pentane Figure 1. Mutual contribution of pairs of vertices (connected by curved lines) to the VC(a,b,c) resp. VNL(a,b,c) indices in pentanes. Dashed line (- - - -): contribution beyond any doubt, but other vertex pairs are to be checked for their inclusion to the VC(a,b,c) or VNL(a,b,c) indices Dashed-dotted line (- ? - ? - ? -): contribution of vertex pairs either equivalent to those marked with the dashed line to the VC(a,b,c) indices or those not equivalent to them to VNL(a,b,c) indices on grounds of structural similarity or dissimilarity Dotted line (? ? ? ? ? ?): contribution to the VC(a,b,c) or VNL(a,b,c) indices based on the distance from the diagonal element 2-2 in the matrix degree mentioned above, combining it with the idea of structural interpretation using the contribution of interior and terminal vertices34-36 of the alkane graph, followed the idea: 1. Let us join the mutual contributions of pairs of interior vertices of equal degree, 2. Let us join the mutual contributions of pairs of interior vertices of different degree, 3. Let us join the mutual contributions of all vertices of different degree, which are not considered under point 2, 4. Let us join the mutual contributions of pairs of terminal vertices of degree one. Doing it, we have four additional groups of indices, for which the mutual contribution of graph vertices to their value is indicated in Figure 2 for the structure of the 2,3-dimethyl pentane: A(a,b,c): (Sum of matrix elements having vi = vj > 1)left B(a,b,c): (Sum of matrix elements having vi ? vj where (vi and vj) > 1)left Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix 182 Acta Chim. Slov. 2006, 53, 180–190 C(a,b,c): (Sum of matrix elements having vi = 1 and vj > 1, or vice versa)left D(a,b,c): (Sum of matrix elements having vi = vj = 1)left These indices can be combined to give additional groups of indices: N-\ AB(a,b,c) AC(a,b,c) AD(a,b,c) BC(a,b,c) BD(a,b,c) CD(a,b,c) ABC(a,b,c) ABD(a,b,c) ACD(a,b,c) BCD(a,b,c) ABCD(a,b,c) = A(a,b,c) = A(a,b,c) — A(a,b,c) = B(a,b,c) = B(a,b,c) = C(a,b,c) = A(a,b,c) = A(a,b,c) = A(a,b,c) = B(a,b,c) = A(a,b,c) D(a,b,c) + B(a,b,c) + C(a,b,c) + D(a,b,c) + C(a,b,c) + D(a,b,c) + D(a,b,c) + B(a,b,c) + B(a,b,c) + C(a,b,c) + C(a,b,c) + B(a,b,c) + C(a,b,c) + D(a,b,c) + D(a,b,c) + D(a,b,c) + C(a,b,c) VL(a,b,c) (ref.1) VC(a,b,c) \ I 2—3----4—5 1 A(a,b,c) VNL(a,b,c) 7 I ¦3: ¦4—5 B(a,b,c) 4—-5 C(a,b,c) D(a,b,c) Figure 2. Mutual contribution of pairs of vertices (connected by curved lines) to the indices VC(a,b,c), VNL(a,b,c), A(a,b,c), B(a,b,c), C(a,b,c), and D(a,b,c) in 2,3-dimethyl pentane (23M5). The A(a,b,c), B(a,b,c), C(a,b,c), and D(a,b,c) indices of n-alkanes can be expressed analytically: N-l A(a,b,c)n =2a+ ^ (N-y)(y-2)c y=3 B(a,b,c)n = 0 C(a,b,c)„ = (2a+2 ) y*; (N-y)c D(a,b,c)n = (N-1)c 0 3a 0 3a 2c 3a3b 0 2a 3c 2a3b 2c 2a3b 0 4C 3b 3C 5*2C 2* 0 2c 3t 5*2C 2*5C 4C 0 3c 3i ^ 3b 2*2C 3C 3C 0 Figure 3. Left half of the universal (Dval14,24) matrix of 2,3-dimethylpentane in the (a,b,c) notation. Bold: Matrix elements contributing to the A(a,b,c) indices Bold: Matrix elements contributing to the B(a,b,c) indices Italics: Matrix elements contributing to the C(a,b,c) indices Normal script: Matrix elements contributing to the D(a,b,c) indices Matrix elements, which at the 2,3-dimethylpentane structure contribute to the VC(a,b,c) resp. VNL(a,b,c) indices were presented previously.1 Those ones which contribute to the A(a,b,c), B(a,b,c), C(a,b,c), resp. D(a,b,c) indices are presented in Figure 3, from which can be deduced also the combinations that contribute to additional groups of indices defined above. At particular structures, the value of some indices is equal to zero: The indices A(a,b,c) of propane (Pr), 2M3, 2M4, 22M3, 22M4, 223M4, 224M5, 223M5, and 233M5, as well as the indices B(a,b,c) of n-alkanes, 2M3, 22M3, 23M4, 234M5, and 2233M4. An infinite number of reverse analogues of the above-mentioned indices [BRI(a,b,c) and IRI(a,b,c) indices] can be derived from the universal matrix using the Reverse Distance Matrix38 approach of Balaban et al.19 where d = dmax - dij, which in our case becomes BRdij = dmax - dij, and the Complementary Distance Matrix approach of Ivanciuc et al.20 where d = dmax + dmin - dij, which in our case becomes IRdij = 1 + dmax - dij,. Some of the newly derived indices present a clear structural characteristic: Indices AB(a,b,c) represent the mutual contributions of all interior vertices. Indices C(a,b,c) represent the mutual contributions of vertex pairs composed of an interior and a terminal vertex. The D(a,b,c) indices represent the mutual contributions of all terminal vertices. Thusly, the indices AB(a,b,c), C(a,b,c), and D(a,b,c) can be used for this type of structural interpretation of other indices derived from the left half of the universal matrix. For indices derived from the whole universal matrix by summation of its nondiagonal elements, the indices AB(a,b,c) + AB(b,a,c), C(a,b,c) + C(b,a,c), and 2×D(a,b,c) can be used. Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix Acta Chim. Slov. 2006, 53, 180–190 183 On the other hand, the AD(a,b,c) indices can be used to assess some symmetric aspects of other indices derived from the left half of the universal matrix, whereas the BC(a,b,c) indices can be used to assess some asymmetric aspects of other indices derived from the left half of the universal matrix. The AV(a,b,c) indices1,7 (for whole-matrix indices like V(a,b,c);1 whereas 1/2AV(a,b,c) for half-matrix indices like VL(a,b,c),1), can be used to assess the anti-symmetric aspects of other indices derived from the universal matrix, but only when the exponent a ^ b, which is not the case among many well-known indices, since they are derived from symmetric matrices. From the universal matrix it is possible to derive also the indices, which represent a special case in structural interpretation, i.e. the concept of mutual contribution of vertices forming the terminal bonds.34,35 These are the C(a,b,-oo) indices. Structures of all n-alkanes give rise to equal values of the terminal bond indices; therefore these indices are useful for the structural interpretation of indices of alkanes of the same size. However, while the C(a,b,-oo) indices represent the mutual contribution of vertices forming the terminal bonds, the ABCD(a,b,-oo) (= VL(a,b, -oo),1) indices represent the mutual contribution of all adjacent vertices, i.e. of all those forming the CC bonds in the alkane in question. Consequently, the difference ABCD(a,b,c) - ABCD(a,b,-oo) gives the mutual contribution of the non-adjacent vertices, whereas the difference ABCD(a,b,c) - C(a,b,-oo) gives the mutual contribution of all the vertices which do not form the terminal bonds. The reverse analogues of terminal bond indices are IRT(a,b,c) = C(a,b, -°°)*(dmax - 1)c, as well as BRT(a,b,c) = C(a,b, -°o)*(dmax - 2)c, where dmax = N - Np + 1; N is the carbon number of the alkane in question and N is the number of primary carbons in its structure (i.e. the number of vertices of degree one in its graph). In a similar way as the summation-derived indices considered above, also the multiplication-derived indices can be derived and used. If we define: P[A](a,b,c): (Product of elements having vi = v > 1)left P[B](a,b,c): (Product of elements having vi ^ v where (vi and v) > 1)left P[C](a,b,c): (Product of elements having vi = 1 and vj > 1, or vice versa)left P[D](a,b,c): (Product of elements having vi = vj = 1)left Then, P[AB](a,b,c) = P[A](a,b,c) × P[B](a,b,c) P[ABCD](a,b,c) = P[A](a,b,c) × P[B](a,b,c) × P[C](a,b,c) × P[D](a,b,c) [= PL(a,b,c), ref.1] and the fractions of contributions interesting for interpretation are P[AB](a,b,c)/(P[ABCD](a,b,c))1/3, P[C](a,b,c)/(P[ABCD](a,b,c))1/3, and P[D](a,b,c)/P[ABCD](a,b,c))1/3. Their product is equal to one. Usefulness of new indices The usefulness of indices derived from parts of the universal matrix by summation of the corresponding elements, based on data of 29 physicochemical properties of alkanes from propane to octanes inclusive is presented in Table 1 and 2, whereas the usefulness based on 31 physicochemical properties of octanes is presented in Table 3 and 4, where the absolute value of the highest observed correlation coefficient, abs(Rfit)max, of linear relationship of that type of index to the physicochemical property in question is indicated as a guide. The index giving rise to abs(Rfit)max > 0.99 accounts for > 98% of variance in the data set. Among the physicochemical properties MR, Tc/Pc, and ?, the correlations of best ABC(a,b,c) indices are equal to those of best VL(a,b,c) indices. In both types of indices they appear at the same values of exponents a, b, and c. From this fact we can conclude that in these cases the mutual contribution of terminal vertices doesn’t contribute any important information. Among BP/Tc, B P, ?Hv, ?Hf°g, Tc2/Pc, and logVP, the correlations of the best ABC(a,b,c) indices are better than those of the best VL(a,b,c) indices. In these cases, the mutual contribution of terminal vertices deteriorates the usefulness of the index. At Vc, the correlation of the best ABC(a,b,c) index is slightly worse than that of the best VL(a,b,c) index. Interesting is also that an AB(a,b,c) index indexes the best the surface tension of alkanes (ST), but only to Rfit = 0.975, which is worse than Rfit = -0.986 of V2/N(-?, 4.8, 3.0).1 The situation is quite different considering only octanes, Table 3 and 4. Besides some VL(a,b,c) indices, also some indices of the groups C(a,b,c), AB(a,b,c), BC(a,b,c), CD(a,b,c), ABC(a,b,c), and ABD(a,b,c) give rise to abs(Rfit)max > 0.99. This observation differs from that in the previous paper,1 where among topological indices derived from the whole universal matrix or from its halves there was as a rule of thumb, abs(Rfit)max at n-alkanes >> all alkanes from propane to octanes inclusive > octanes. Interesting is that most of the best members of these new index groups are the best indices of Octane Numbers (BON, MON, and/or RON), of BP/Tc and Pc, but not of Tc/Pc or Tc2/Pc. The best of them are BC(1.40, -0.50, 2.6) correlating to BON with Rfit = -0.997, ABC(1.14, 1.15, -3.1) correlating to BP/Tc with Rfit = -0.997, AB(1.41, 1.14, 0.53) correlating to RON with Rfit = 0.995, AB(1.50, 0.96, 1.42) correlating to MON with Rfit = 0.994, and ABD(1.83, 1.84, 1.68) correlating to Pc with Rfit = 0.994. Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix 184 Acta Chim. Slov. 2006, 53, 180–190 Table 1. Best correlations of indices derived from parts of the universal matrix with tested physicochemical properties of alkanes from propane to octanes inclusive. abs(i?ffi)max 0.90 —> 0.95 0.95—> 0.99 > 0.99 Index ViXajbjC),1 = nD, ST, d, RON ac , C, B, Tc, Vm logVP, Vc, <», BP/Tc, AHf°g, ABCD(a,b,c) BP, AHv, Tc2/Pc, Tc/Pc, MR VC(a,b,c) MON, nD, Pc, BON, ac, C, d, RON, to, BP, AHv, AHf g, logVP, MR, Vc, BP/Tc, Vm, B, Tc Tc/Pc, Tc2/Pc A(a,b,c) BON, RON, MON C(a,b,c) MON, ac, RON, BON, Pc, B, C Tc, BP/Tc, BP, AHv, logVP, Vc, Vm, AHf g, Tc/Pc, Tc2/Pc, to, MR D(a,b,c) <», BP/Tc, Vm, AHf°g, MR, Tc2/Pc, Vc, Tc/Pc AB(a,b,c) ac, AHf g, C, Vm, BP/Tc, nD, d Vc, Tc, BP, MON, logVP, Tc/Pc, to, MR, B, BON, AHv, RON, Tc2/Pc, ST AC(a,b,c) ST, BON, nD, d, Pc, ac, C BP/Tc, Vm, B, to, Tc, logVP, Vc, BP, AHv, Tc/Pc, MR Tc2/Pc AD(a,b,c) AHv, BP, <», logVP, Vm, BP/Tc, AHf°g, Tc/Pc, Vc, Tc2/Pc, MR BC(a,b,c) BON, C, Vm, Vc, BP/Tc, B, Tc/Pc Tc, MR, AHv,AHf g, logVP, BP, co, Tc2/Pc BD(a,b,c) BP/Tc, <», Vm, MR, AHf°g, Tc2/Pc, Vc, Tc/Pc CD(a,b,c) Pc, d, ac, C BP/Tc, <», B, Tc, Vm, logVP, BP, AHv, AHf g, Vc, Tc/Pc, MR, Tc2/Pc ABC(a,b,c) V", BON, ST, nD, d, ac C, B, Pc, Vm, Tc logVP, Vc, <», BP/Tc, BP, AHv, AHf°g, Tc2/Pc, MR, Tc/Pc ABD(a,b,c) MON, Pc, Vm, nD, RON, ac, d C, AHf g, BP/Tc, BON, Tc, Vc, BP, MR, B, <», logVP, Tc/Pc, ST, AHv, Tc2/Pc ACD(a,b,c) nD, Pc, d, ac, C, BP/Tc, to Vm, B, Tc, BP, AHv, logVP, AHf g, Vc, Tc/Pc, MR, Tc2/Pc BCD(a,b,c) Pc, d, ac, C BP/Tc, B, <», Vm, Tc, AHv, BP, logVP, AHf g, Vc, Tc/Pc, MR, Tc2/Pc B(a,b,c): No abs(Rfit) > 0.90 has been observed. AB(a,b,c): The index does not exist at Pr, 2M3, and 22M3. Therefore, its value at these alkanes was set to be equal to zero. Bold: Better than the best respective indices derived from the reverse types of universal matrix. Table 2. Best correlations of indices considered here with tested physicochemical properties of alkanes from propane to octanes inclusive having abs(Rfit)max > 0.99; (VL(a,b,c) ? ABCD(a,b,c)). PP I(a,b,c) N Rfit Rcv Sfit Scv F Tc/Pc ABC(0.86, -0.40, -°») 38 0.9991 0.9990 1.48 1.61 21038 Vl (0.86, -0.40, -°o) 38 0.9991 0.9990 1.48 1.61 21038 MR ABC(1.06, -0.093, -°») 38 0.9988 0.9983 0.304 0.356 15089 Vl (1.06, -0.093, -°») 38 0.9988 0.9983 0.304 0.356 15089 Tc /Pc ABC(0.182, -0.170, -°») 38 0.9981 0.9975 1583 1799 9537 Vl (-0.0109, -0.46, -3.4) 38 0.9981 0.9976 1601 1801 9351 AHfg ABC(1.61, -0.33, -4.3) 38 0.9974 0.9970 0.525 0.555 6825 Vl (-0.28, 0.136, -°») 38 0.9934 0.9925 0.830 0.879 2710 AHv ABC(-0.49, -0.48, -°») 38 0.9969 0.9961 0.074 0.079 5743 Vl (-0.50, -0.59, -5.09) 38 0.9966 0.9961 0.077 0.082 5330 BP ABC(-0.50, -0.37, -6.2) 38 0.9950 0.9940 4.19 4.53 3574 Vl (-0.61, -0.54, -4.7) 38 0.9950 0.9894 4.19 4.56 3559 BP/Tc ABC(0.62, -0.92, -°») 38 0.9945 0.9907 0.0018 0.0020 3254 Vl (0.62, -0.92, -°») 38 0.9945 0.9907 0.0018 0.0020 3254 CO ABC(0.32, -1.12, -°o) 38 0.9936 0.9931 0.0066 0.0068 2789 Vl (0.32, -1.12, -°») 38 0.9936 0.9931 0.0066 0.0068 2789 Vc ABC(1.00, -0.34, -°o) 38 0.9924 0.9917 0.0090 0.0093 2339 VL (0.98, -0.38, -5.1) 38 0.9924 0.9917 0.0090 0.0093 2346 logVP ABC(-0.52, -0.44, -°») 37 -0.9905 0.9891 0.0941 0.0993 1819 Vl (-0.55, -0.50, -6.1) 37 -0.9905 0.9892 0.0944 0.0996 1811 =, >, and <, meaning equal to, better than resp. worse than, refer to the absolute values of non-rounded data relative to the half-matrix index VL(a,b,c) presented in this Table. Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix Acta Chim. Slov. 2006, 53, 180–190 185 Table 3. Best correlations of indices derived from parts of the universal matrix with tested physicochemical properties of octanes. abs(i?fit)max 0.90 —> 0.95 0.95 —> 0.99 > 0.99 Index ViXa,b,c),' = BP, ST, AHPg, V", Vm AHv, R2, d, nD, S, Pc, C, Tc2/Pc, MON, RON, BP/Tc, ABCD(a,b,c) BON to, Tc/Pc VC(a,b,c) BP/Tc, Vm, V", Pc, C, co, Tc2/Pc, d, RON, BON, Tc/Pc, nD, MON VNL(a,b,c) AHPg, BP, BP/Tc, S, AHv, R MON, co, Tc /Pc, BON, C, RON A(a,b,c) Vm, V", RON, R2, BON, MON, d nD C(a,b,c) ST, BP, Pc AHPg, AHv, R2, S, C, BP/Tc, to, RON, Tc/Pc, MON, Tc /Pc BON D(a,b,c) R , BP/Tc, S, C, Pc, to, BON, Tc /Pc, RON Tc/Pc, MON AB(a,b,c) BP, AHPg, AHv, ST, C, S, V", d, R2, nD, to, BP/Tc, Tc/Pc, Tc2/Pc, BON, RON Vm Pc, MON AC(a,b,c) ac, ST, V", Vm, BP, AHPg, d, S, C, R2, nD, co, BP/Tc, Tc2/Pc, AHv, Pc Tc/Pc, BON, RON, MON AD(a,b,c) R , BP/Tc, V", Vm, S, C, Pc, d, to, BON, Tc /Pc, RON Tc/Pc, MON, nD BC(a,b,c) ST, Pc, BP, AHPg, AHv S, BP/Tc, R2, Tc/Pc, C, co, Tc2/Pc, RON MON, BON BD(a,b,c) R2, BP/Tc, AHv, Pc, MON, S C, Tc/Pc, BON, to, Tc2/Pc, RON CD(a,b,c) ST, BP R , AHPg, AHv, Pc, S, C, BP/Tc, MON, Tc/Pc, to, Tc2/Pc, BON RON ABC(a,b,c) BP, ST, AHPg, V", Vm AHv, d, R2, S, nD, Pc, C, Tc2/Pc BON, MON, RON, to, Tc/Pc, BP/Tc ABD(a,b,c) BP, AHPg, AHv, ST, S, C V", Vm, R2, d, nD, to, BP/Tc, Tc/Pc, BON, MON, Pc, Tc /Pc RON ACD(a,b,c) ST, V", Vm, BP, AHPg, d, Pc, S, nD, C, Tc/Pc, Tc /Pc, co, BON, RON AHv, R , BP/Tc MON BCD(a,b,c) ST, BP, AHPg, Pc AHv, R2, S, C, BP/Tc, co, Tc/Pc, Tc2/Pc, MON, BON RON B(a,b,c): No abs(Rfit) > 0.90 has been observed. Bold: Better than the best respective indices derived from the reverse types of universal matrix. Table 4. Best correlations of indices considered here with tested physicochemical properties of octanes having abs(Rfit)max > 0.99. PP I(a,b,c) N Rfit Rcv Sfit Scv F BON BC(1.40, -0.50, 2.6) 17 -0.9973 0.9971 2.52 2.60 2790 > C(1.07, -0.060, 2.3) 17 0.9956 0.9943 3.23 3.68 1687 > ABC(0.77, 0.35, 1.45) 17 -0.9944 0.9926 3.64 4.18 1326 > AB(1.30, 1.13, 0.74) 17 0.9924 0.9901 4.24 4.84 974 > ABD(1.30, 1.24, 0.42) 17 0.9902 0.9874 4.81 5.41 754 > VL(0.69, 0.150, 1.03) 17 -0.9884 0.9852 5.24 5.86 633 MON C(1.61, 0.36, 3.0) 17 0.9943 0.9927 3.60 4.11 1307 > ABC(1.43, 0.61, 2.1) 17 -0.9941 0.9925 3.65 4.17 1268 > AB(1.50, 0.96, 1.42) 17 0.9940 0.9922 3.68 4.25 1248 > ABD(1.17, 0.94, 0.92) 17 0.9934 0.9915 3.86 4.42 1133 > BC(1.83, -0.37, 3.2) 17 -0.9933 0.9921 3.89 4.24 1115 > VL(1.12, 0.109, 2.1) 17 -0.9928 0.9906 4.06 4.65 1025 CD(1.58, -0.64, 3.2) 17 -0.9903 0.9875 4.70 5.37 761 < Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix 186 Acta Chim. Slov. 2006, 53, 180–190 Table 4. Continued RON IRCD(0.32, -1.44, 0.160) 17 0.9951 0.9939 3.63 4.05 1530 > AB(1.41, 1.14, 0.53) 17 0.9950 0.9934 3.69 4.25 1475 > CD(-3.2, -1.12, -3.2) 17 0.9941 0.9929 3.99 4.36 1260 > VL(-2.4, -0.67, -3.8) 17 0.9940 0.9926 4.04 4.44 1229 ABD(1.40, 1.28, 0.31) 17 0.9938 0.9920 4.08 4.68 1206 < ABC(1.03, 0.45, 1.09) 17 -0.9934 0.9918 4.22 4.71 1125 < BRCD(1.99, 0.71, 0.040) 17 0.9929 0.9909 4.39 4.96 1042 < C(1.37, 0.71, -0.42) 17 0.9923 0.9898 4.57 5.28 958 < BP/Tc ABC(1.14, 1.15, -3.1) 18 -0.9965 0.9956 0.00073 0.00082 2290 > VL(1.15, 1.10, -2.8) 18 -0.9964 0.9955 0.00074 0.00083 2233 Tc/Pc Vl(1.10, 1.02, 1.05) ABCQ.39, 1.35, -0.82) ABC(1.45, 1.47, 0.40) C(1.33, 1.15, 1.98) CD(0.52, 0.00, 2.1) 18 18 18 18 18 -0.9984 -0.9978 -0.9976 0.9922 0.9907 0.9979 0.9972 0.9971 0.9900 0.9879 0.47 0.55 0.57 1.03 1.13 0.54 0.62 0.63 1.15 1.27 4951 3614 < 3329 < 1020 < 847 < CO Vl(-0.108, -0.22, -1.20) 18 -0.9973 0.9966 0.0028 0.0031 2914 ABC(0.46, 0.76, 3.8) 18 -0.9960 0.9951 0.0034 0.0037 1969 < ABC(0.42, 0.52, -4.8) 18 -0.9959 0.9950 0.0034 0.0038 1942 < Pc ABD(1.83, 1.84, 1.68) VL(1.32, 0.67, -1.63) 0.9937 0.9733 0.9919 0.9641 0.14 0.29 0.16 0.32 1259 288 > resp. <, meaning better than resp. worse than, refer to the absolute values of non-rounded data relative to the half-matrix index VL(a,b,c) presented in this Table. Considering only octanes, there perform the best those indices, which present a clear structural characteristic. Among them, the best ones of the indices AB(a,b,c) (presenting only mutual contributions of all interior vertices) and the best ones of the indices C(a,b,c) (presenting contributions of vertex pairs composed of an interior and a terminal vertex) perform better than the best ones of the indices D(a,b,c), which present only the mutual contributions of terminal vertices. Cross-validation supports the validity of results presented in Table 2 and 4. Some members of groups of these indices are thus promising indices of some physicochemical properties of alkanes. Considering the Octane Numbers, Table 5, they give rise to better results than the three-parameter correlations of Hosoya.39 Table 5. Best correlations of indices considered here with Octane Numbers of heptanes (C7) and octanes (C8). ON I(a,b,c) C7 C Rfit S fit Rfit S fit BON BC(2.1, -0.057, 2.8) -0.998 2.2 -0.992 4.2 BC(1.40, -0.50, 2.6) -0.996 3.2 -0.997 2.5 C(1.61, 0.48, 2.5) -0.996 3.1 -0.992 4.3 C(1.07, -0.060, 2.3) -0.994 3.8 -0.996 3.2 MON BC (1.92, -0.26, 3.3) -0.996 3.0 -0.993 3.9 BC (1.83, -0.37, 3.3) -0.996 3.1 -0.993 3.9 AB(1.17, 0.65, 1.59) -0.992 4.3 -0.991 4.6 AB(1.50, 0.96, 1.42) -0.990 4.9 -0.994 3.7 RON AB(1.27, 0.79, 0.98) -0.990 5.0 -0.991 5.0 AB(1.44, 1.17, 0.51) -0.983 6.6 -0.995 3.7 Using the best indices in Table 5 we estimated the missing data of Octane Numbers in the set in use. RON of 2,3-dimethyl butane (23M4) is estimated to be 92.1. For 2,2,3,3-tetramethyl butane (2233M4), the estimation gives the following values: BON = 117.4 and 122.9; MON = 116.0 and 117.2; RON = 137.5. The most promising index derived from the reverse types of the universal matrix, IRCD(0.32, -1.44, 0.160), gives rise to RON = 139. Values estimated by extrapolation using the best observed indices have to be checked by independent approaches. In present case few of them are available. On the one hand, RON values of 2233M4 estimated here are comparable to RON = 135 estimated by Hosoya.39 On the other hand, experimental29 BON values of 2233M5 and 2233M6 are similar to BON of 2233M4 estimated here. Thusly, our estimated values seem to be quite close to the real ones. The pattern of Octane Numbers, where the estimated values for 2,2,3,3-tetramethylbutane are also included, is presented in Figure 4 for heptanes and octanes. Considering all alkanes from propane to octanes inclusive, the best indices derived from the universal (Dval13,24) matrix are in general better than the best ones derived from the reverse types of this matrix. The physicochemical property, for which the best indices derived from the universal matrix are evidently better than the best ones derived from the reverse types of the universal matrix, is the surface tension of alkanes. Considering only octanes, there are the best indices derived from the universal matrix in general better than the best ones derived from the reverse types of the universal matrix, as well. Exceptions are noticed mainly at physicochemical properties C, ?, and S, where 0.95 < abs(Rfit) < 0.99 is observed. Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix Acta Chim. Slov. 2006, 53, 180–190 187 160 140 120 100 80 - 60 40 - 20 -0 - BON C8 - MON C8 - RON C8 - BON C7 - MON C7 - RON C7 -20 • "(^^(^^(^^c^|j0,""0,0,"" Figure 4. The pattern of Octane Numbers (BON, MON, RON) of heptanes and octanes. For 2233M4 the values estimated by new indices considered here (see text) are taken. The correlation of the best indices that contain only the information of the mutual contribution of vertices, which are involved in terminal CC bonds, i.e. of the C(a,b,-?) indices, to the values of physicochemical properties of octanes is remarkable but in all cases abs(Rfit) < 0.99. Noticeable is the fact that all of the best indices of this type correlate the best i.a. with measures of the Octane Number. Estimation of BON of nonanes and decanes The existence of quite good indices of Octane Numbers of octanes, giving rise to (Rfit) = -0.994 to -0.997, Sfit= 3.7 to 2.5, Table 5, is encouraging. Namely, since there do exist good indices of Octane Numbers of octanes, then it can be reasonably expected that there exist also good indices of Octane Numbers of nonanes and decanes and that these good indices will be among those ones, which correlate well with available data. However, there are available only few data of Octane Numbers of nonanes and decanes, i.e. six BON data of nonanes29 as well as of decanes29 and only three RON data of nonanes.39 Therefore, the estimation of Octane Numbers of these groups of alkanes is an extremely risky task. There is also another problem. Besides 2,5,5- and 3,3,5-trimethylheptane, also the values of BON of n-nonane and n-decane are outliers in the plot of Morley29 as well as in comparison with other n-alkanes. Nonanes If, in analogy to n-octane, where the change of value proved successful,39 we estimate the BON value of n-nonane to be not -17 as presented by Morley29 but possibly around -30, which follows from the regression of Morley’s29 k and BON data, or even -38, which follows from the approx. linear decrease of BON values with the size of n-alkane as observed in BON data from n-pentane to n-octane, then we obtain the best indices for nonanes to be those presented in Table 6. The pattern of results obtained using these indices is illustrated in Figure 5 based on BONNon = -30. The best C(a,b,c) indices mentioned in Figure 5 seem to Table 6. The best indices for BON of nonanes. BONNon Index R -1730 -30 -38 ABC(0.96, 2.2, -0.56) ABC(0.72, 1.78, -<*>) VL(1.07, 2.2, -0.48) Vl(-0.51, 1.59, -3.3) C(1.12, 1.30, -0.160) C(0.54, 0.65, -1.13) VL(-1.43, -0.089, -=») C(1.08, 1.26, -0.079) C(-0.33, 0.113, -») VL(-1.52, -0.115, -=») Vl(-°°, -1.29, -3.3) C(1.02, 1.20, -0.084) C(-0.37, 0.088, -<*>) ViX-1.57, -0.128, -») Vl(-°°, -1.20, -3.4) S 0.999614 0.55 0.998528 1.07 0.999516 0.61 0.998604 1.04 0.998385 3.14 0.997915 3.57 0.997951 3.53 0.998523 3.31 0.997610 4.30 0.996665 4.97 0.993622 6.87 0.998587 3.43 0.997230 4.79 0.995749 5.94 0.993028 7.60 none Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix 188 Acta Chim. Slov. 2006, 53, 180–190 140 120 100 80 60 40 20 0 -20 -40 -¦—BON exp. C 1.08 — C -0.33 -X—BC -0.73 —I-----VL -1.52 -------VL -oo ¦•- - ¦ Heptanes -±— Octanes a\erg. <& CONNSNNNSMDtDtDNSMCinCDtDlOIDtDtDlOIDtDffiinminini i^-Jo^cDm^-com^-cNCNi^-CNico ^ CMCMCMCMCOCOLULULUCMCO CO ^J- CO co^om^-^-m^-cocM^-co^-co^-^-^-i LLl?oCOCOCMCMCMCMCM^-COCOCMCOCO^-( JoCMCMLU^^^LU^^COLLJ^COCMI CO CO CO ^ ^ ^ ' Figure 5. Estimated BON values of nonanes, taking BONNon = -30. Indices: C(1.08, 1.26, -0.079), C(-0.33, 0.113, -?), BC(-0.73, 0.74, -1.16), VL(-1.52, -0.115, -?), VL(-?, -1.29, -3.3). present the lower limit of possible BON values. To help judging the upper limit, the BON values of heptanes and octanes having the same type of the branched structure were taken in consideration. Since no octane having the same type of branched structure has a higher BON value than the corresponding heptane, the same rule seems plausible also for nonanes. Thus, the upper limit of plausible BON values of nonanes present the BON values of octanes. To estimate better the Octane Numbers of nonanes, the average of estimated values is taken and it is presented in Table 7. When no value of n-nonane is considered among the input data, only the interpolated results seem promising, whereas those extrapolated towards less branched nonanes are obviously too high, higher than those of octanes since the apparently best indices taken into account do not index all the curvature of the relation of ON vs. structure. As an example: the result for BONNon = 50. Some other indices, correlating quite well to available BON data of nonanes, proved useless since they give rise to an evidently unacceptable scatter of results. One of them is e.g. VL(4.6, 1.89, 0.89), correlating to available data of nonanes with Rfit = 0.9989, Sfit= 3.05, but giving rise to some improbable results such as BON2244M5 = 365, BON244M6 = 208, BON26M7 = 99, BON223M6 = -24, BON2iPr6 = -48. Among decanes a still worse situation is among VL(0.35, 0.00, -4.7), correlating to available data of decanes with Rfit = -0.9995, Sfit= 2.08, but giving rise to BON3E22M6 = 1043, BON22344M5 = -1417, as well as among VL(0.79, 0.00, -5.1), correlating to available data of decanes with Rfit = -0.9939, Sfit= 6.90, but giving rise to BON3E22M6 = 674, BON22344M5 = -1643. Table 7. Estimated BON values of nonanes, derived as the average of values calculated from the indices mentioned in text. BONNon none -17 -30 -38 Nonane BON Non -17 -16 -29 -36 2M8 18 13 6 3M8 20 15 7 4M8 20 10 7 3Et7 19 18 10 4Et7 18 17 9 26M7 51 49 43 25M7 53 54 47 24M7 53 54 48 23M7 54 53 48 35M7 54 55 48 34M7 55 55 48 3E2M6 54 55 51 4E2M6 53 56 50 3E4M6 55 57 51 22M7 56 53 48 33M7 58 55 48 44M7 59 55 48 3E3M6 59 59 53 33Et5 58 61 56 235M6 81 82 85 84 84 234M6 87 87 87 88 3E24M5 88 87 86 85 87 225M6 91 91 87 87 84 224M6 95 88 92 88 223M6 100 90 92 88 3E22M5 108 108 108 107 107 244M6 92 90 92 88 233M6 98 91 92 88 334M6 99 92 93 89 3E23M5 103 92 96 93 2234M5 109 119 120 123 2334M5 109 121 121 123 2244M5 108 120 121 122 2233M5 123 123 125 126 128 Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix Acta Chim. Slov. 2006, 53, 180–190 189 Decanes If, in analogy to n-octane, where the change of value proved successful,39 we estimate the BON value of n-decane to be not -41 as presented by Morley29 but possibly -30, which follows from the regression of Morley’s29 k and BON data, or -57, which follows from the approx. linear decrease of BON values with the size of n-alkane as observed in BON data from n-pentane to n-octane, we obtain the best indices of BON to be those presented in Table 8. Table 8. The best indices to indicate BON values of decanes at supposed values of BONDec. BONDec Index R fit S fit -31 -4130 -57 C(3.0, -1.59, 2.0) AC(3.0, -o«, 2.1) AC(3.6, -0.71, 1.74) BC(3.0, -4.7, 2.0) AC(3.0, -3.2, 2.0) C(2.8, -o«, 2.3) AC(3.0, -2.9, 2.1) AC(3.6, -1.14, 2.0) BC(2.6, -o«, 2.1) AC(3.2, -1.63, 1.92) AC(3.5, -0.99, 1.85) -0.999739 -0.999798 -0.999385 -0.999636 -0.999092 -0.992804 -0.998550 -0.999395 -0.992559 -0.999268 -0.999986 1.12 0.99 1.72 1.32 2.54 7.53 3.38 2.19 7.65 2.62 0.37 The indices C(2.8, -?, 2.3) and BC(2.6, -?, 2.1) seem to give the most plausible results. A similar conclusion holds also in other cases, i.e. when the value of BONDec is omitted as well as when it is taken to be BONDec = -31 or -57. The best C(a,b,c) indices seem on the one hand to present the lower limit of possible BON values and on the other hand to indicate on average the most plausible estimated BON values of decanes. Table 9. Estimated BON values of decanes, means of best results.* BONDec** -31 -41 -57 Decane BON Dec -41 -2 -30 -37 -57 2M9 53 47 37 50 3M9 48 37 30 34 4M9 37 25 20 18 5M9 22 10 5 2 3Et8 52 42 37 39 4Et8 45 36 32 33 4Pr7 42 33 29 32 27M8 20 20 21 14 20 26M8 45 44 38 43 25M8 65 63 58 65 24M8 80 79 72 84 23M8 90 91 83 101 36M8 39 37 33 37 35M8 59 55 52 57 34M8 74 71 66 74 45M8 48 48 44 42 46 5E2M7 45 44 38 51 Table 9. Continued 4E2M7 5E3M7 3E2M7 3E4M7 4E3M7 4iPr7 34Et6 22M8 33M8 44M8 3E3M7 33Et6 4E4M7 236M7 246M7 235M7 245M7 234M7 345M7 3E25M6 4E23M6 3E24M6 3iPr2M6 226M7 225M7 224M7 223M7 255M7 31 244M7 233M7 335M7 77 334M7 344M7 4E22M6 3E22M6 4E33M6 4E24M6 3E23M6 3E34M6 33E2M5 3iPr24M5 2345M6 2245M6 2235M6 2234M6 2335M6 2344M6 2334M6 3E224M5 3E234M5 2255M6 2244M6 2233M6 126 3344M6 3E223M5 22344M5 22334M5 70 47 85 76 70 18 71 94 76 45 79 80 50 70 60 90 79 105 73 83 84 98 44 74 94 109 119 32 76 107 75 91 70 82 105 76 63 101 91 90 62 95 98 109 124 97 91 112 107 103 51 96 127 77 110 102 123 69 45 84 73 68 17 68 92 71 39 74 76 45 71 60 91 80 106 73 83 84 98 45 77 95 110 121 33 77 109 75 90 69 82 105 75 63 100 91 89 62 95 100 111 126 98 93 113 108 104 52 96 128 77 110 103 124 64 41 78 70 65 18 67 86 70 42 74 77 49 68 58 88 78 103 72 81 81 96 48 73 92 106 116 38 79 106 76 91 73 78 102 76 65 100 92 90 68 95 100 110 124 100 97 114 107 105 59 100 128 85 111 109 126 76 50 92 76 72 22 73 95 68 32 72 76 40 64 54 89 78 110 75 83 92 102 50 84 100 114 127 32 77 113 77 90 67 92 112 78 67 104 91 95 56 89 104 112 129 94 91 112 114 104 37 88 127 70 112 98 121 * Average values obtained using indices: **BONDec = none: C(3.0, -1.59, 2.0), AC(3.0, -?, 2.1), BC(3.0, -4.7, 2.0) **BONDec = -31: AC(3.0, -3.2, 2.0) **BONDec = -41: C(2.8, -?, 2.3), AC(3.0, -2.9, 2.1), BC(2.6, -?, 2.1) **BONDec = -57: AC(3.2, -1.63, 1.92), AC(3.5, -0.99, 1.85) ** Assumed BONDec none none Perdih and Perdih Topological Indices Derived From Parts of a Universal Matrix 190 Acta Chim. Slov. 2006, 53, 180–190 Using these indices, we get the estimation of BON values of decanes, which is presented in Table 9. Having Table 7 and 9, let us wait for additional experimental data to check the validity of derived values. Comparison of known ON data based on structural features of alkanes, however, indicates that several values estimated above are plausible.40 Conclusions Structural interpretation of indices does give better understanding of the background of their characteristics. The new approach to their structural interpretation using the concept of mutual contribution of vertices proved useful for indices derived from parts of halves of a universal (Dval13,24) matrix. It gave rise to new ideas how to develop new indices from this matrix. Some of these new indices proved useful for structural interpretation of other (e.g. known) indices derived from this matrix, whereas several ones proved to be useful indices of a number of physicochemical properties of alkanes, especially of Octane Numbers. References 1. A. Perdih, B. Perdih, Acta Chim. Slov. 2004, 51, 598-609. 2. A. Perdih, B. Perdih, Acta Chim. Slov. 2002, 49, 67-110. 3. A. Perdih, B. Perdih, Acta Chim. Slov. 2002, 49, 291-308. 4. A. Perdih, B. Perdih, Acta Chim. Slov. 2002, 49, 309-330. 5. A. Perdih, B. Perdih, Acta Chim. Slov. 2002, 49, 467-482. 6. A. Perdih, B. Perdih, Acta Chim. Slov. 2002, 49, 497-514. 7. A. Perdih, B. Perdih, Acta Chim. Slov. 2003, 50, 83-94. 8. A. Perdih, B. Perdih, Acta Chim. Slov. 2003, 50, 95-114. 9. A. Perdih, B. Perdih, Acta Chim. Slov. 2003, 50, 161-184. 10. A. Perdih, B. Perdih, Acta Chim. Slov. 2003, 50, 513-538. 11. H. Wiener, J. Am. Chem. Soc. 1947, 69, 17-20. 12. M. Randić, J. Am. Chem. Soc. 1975, 97, 6609-6615. 13. O. 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