Acta Chim. Slov. 2003, 50, 95-114. 95 SOME TOPOLOGICAL INDICES DERIVED FROM THE vmdn MATRIX. PART 7. THE Vij(m,n) INDICES Anton Perdih,* Branislav Perdih Mala vas 12, SI-1000 Ljubljana, Slovenia Received 20-05-2002 Abstract The best correlations of some Vij(m,n) indices using data from propane through ali octanes are observed at Mw (r = 1), MR (r = 0.998), Tc2/Pc (r = 0.998), AHv (r = 0.997), BP (r = 0.994), Tc/Pc (r = 0.994), AHf°g (r = 0.991), and logVP (r = -0.990). The best correlations using only data of octanes are observed at Tc/Pc (r = -0.998), co (r = -0.996), BP/Tc (r = -0.995), MON (r = -0.987), Tc2/Pc (r = 0.984), C (r = 0.969), S (r = -0.961), and Pc (r = 0.950). The index Vij(-2,-2) has a regular sequence of isomers due to the increase of the size of the molecule. The indices Vij(-V4,-6), Vij(-1,74), Vij(-2,V4), Vij(-2,V3), Vij(3,-74) as well as a group of Vij(m,n) indices having -1 < m < -V4 and -1 < n < -V4 have a regular sequence of isomers due to increasing branching. These indices seem to be good sources of the susceptibilitv for branching derived BIA type branching indices. Introduction Several hundred topological indices have been developed and tested for their performance as branching indices or indices of substances’ properties.1'2 A substantial part of them is derived from one or another matrix associated with molecular structure. Estrada3 developed a matrix that enables the derivation of an infmite number of indices. We4 presented some types of matrices that enable the derivation of an infmite number of indices, too, and we have shown that these matrices represent a step in unification of several matrices used to derive topological indices, i.e. of the adjacency matrix, the distance matrix, the reciprocal distance matrix, etc. Topological indices have been correlated with several physical, chemical, and biological properties of molecules. However, even several properties of alkanes stili cannot be well described with particular available indices5 and combinations of them are to be used.6 In spite of that, interest in topological indices has grown remarkably during recent years. Therefore, the study of those topological indices that might be good branching indices remains an important area of research. In recent papers4'7"11 we studied the characteristics of some groups of indices derived with help of the generalized vertex-degree, vertex distance matrices: the A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 96 Acta Chim. Slov. 2003, 50, 95-114. summation derived W(m,n) indices4 and the "mean degree of vertices" indices,7 the largest eigenvalues of the same matrices,8,9 the susceptibilities for branching of the W(m,n) indices,10 as well as the difference derived indices.11 In present paper are studied the summation derived Vij(m,n) indices. Data and notations The structures of alkanes are presented in shorthand. «-Alkanes are presented in a different way than branched ones, e.g. Hp is «-heptane, Oct is «-octane, whereas 223M5 is 2,2,3-trimethylpentane, 3E2M5 is 3-ethyl-2-methylpentane, etc. Other terms are explained on 2,2-, 2,3- and 2,5-dimethyl hexane (22M6, 23M6 and 25M6) as examples. Ali of them have eight carbons (N = 8) and four of them are primary carbons (Np = 4). The two branches (i.e. the number of branches, Nbr = 2) in 22M6 are positioned on a quaternary carbon (q) placed on the periphery (per) of the molecule. The two branches in 23M6 and 25 M6 are positioned on tertiary carbons (0- In 23M6 the branches are adjacent (adj) and those in in 25M6 are distant (dist). The branches on carbons No. 2 and 5 are placed on the periphery of the molecule, and the one on carbon No. 3 is placed near the centre (ctr) of the molecule. The physicochemical properties The data for the boiling point (BP), density (d), the critical data Te, Pc, Ve, Zc, ac, and de, as well as 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 (co) and the refraetive index (nD) were taken from the CRC Handbook12 or from Lange's Handbook13. The data for the liquid molar volume (Vm), the ratios Tc2/Pc and Tc/Pc used instead of the van der Waals parameters a0 and b0, the ratio BP/Tc (reduced BP), and the molar refraetion (MR) were calculated from data presented in the handbooks. The data for Motor Octane Numbers (MON) was taken from Pogliani14 and Gutman et al.,15 those for vapour pressure (logVP) from Goli and Jurs,16 and those for the entropy (S) and quadratic mean radius (R2) from Ren.17 A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 97 Methods Susceptibility for the increase in carbon number; Susceptibility far branching The susceptibility18 is defined as the normalised difference of the indices' or properties' values, Eq. 1, Sa,b = Xb/Xa-l (1) where Sa,b is the susceptibility, X is an index or a property, subscript a refers to the less branched structure and subscript b refers to the more branched structure. Which one is more branched is concluded by intuition as presented in ref.19 as well as by the Methane based definition and the n-Alkane based definition.20 Two groups of susceptibilities are used. In the susceptibility for the increase in carbon number, denoted as Sa,a+i, subscript a refers mostly to the structure having the same number and type of branches if not explicitly shown othenvise. In these cases, the two alkanes taken into account differ in carbon number by one. For example, in S7;8 the digit 8 means any octane having the same number and type of branches as a heptane which is represented by the digit 7. In Shp,oct the data of «-heptane (Hp) and «-octane (Oct) is used. In S2m6,2m7 the data of 2-methyl hexane (2M6) and 2-methyl heptane (2M7) is used, in S2m6,3m7 the data of 2-methyl hexane (2M6) and 3-methyl heptane (3M7), etc. In the susceptibility for branching, S„,b, the subscript n refers to the «-alkane and the subscript b refers to any alkane of the same carbon number. For example, in SUp,b Hp refers to «-heptane and b to any heptane. In Shp,2m6, 2M6 refers to 2-methyl hexane. Contribution of structural features The relative contribution of structural features to the value of the index is estimated using the susceptibilities. The relative contribution of a structural feature to the value of the index when the size of the molecule increases, is estimated as follows. The contribution of the number of branches is estimated by S33m5,33m6 - S3m7,3m6. The contribution of the position of branches is estimated by 1/2(S33m5,33m6 - S22m5,22m6). The contribution of the separation between branches is estimated by S24m5,24m6 - S23m5,23m6. The contribution of the change of the substituent from methyl to ethyl is estimated by S3e5,3e6 - S3m6,3m7. A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7... 98 Acta Chim. Slov. 2003, 50, 95-114. The relative contribution of a structural feature to the value of the index when the branching of the molecule increases is estimated as follows. The contribution of the number of branches is estimated by S0ct,33M6 - S0ct,3M6. The contribution of the position of branches is estimated by S0ct,34M6 - S0ct,23M6. The contribution of the separation between branches is estimated by S0ct,24M6 - S0ct,23M6. The contribution of the change of the substituent from methyl to ethyl is estimated by S0ct,3E2M5 - S0ct,23M6. The contributions are labelled with letters b, c, s, and e, respectively. The uppercase letter is used to label the structural feature having the highest contribution to the value of index in question. The results of estimations of the relative contribution of structural features are checked by the sequences of isomers obtained by sorting S7;8 or S8>6. Results and discussion The Vij(m,n) indices and the source matrix The Vij(m,n) indices are derived by summation of ali elements of the matrix having the main diagonal elements gu = 0 and the nondiagonal elements g,j (frj) = v/-mxvymxd(,-n, where v, and v,- is the degree of vertex i and j, respectively, (in alkanes it is the number of C-C bonds the carbon in question is involved in) and d!y is the shortest distance from vertex i to vertexy (in alkanes it is the smallest number of bonds between the carbons in question), cf ref4 The Vij(m,n) indices of methane are equal to zero since gu = 0 by defmition. The Vij(m,n) indices of ethane are equal to 2 since lmxlmxln + lmxlmxln is in any čase equal to 2. The values of the index Vij(-00,-00) of other alkanes are equal to zero. Several indices of this group have a long tradition, e.g. the Wiener index,21 W = V2Vij(0,l), the reciprocal Wiener index,22 RW = V2Vij(0,-1), the Harary index,23 H = V2Vij(0,-2), the Zagreb index,24 M = V2Vij(2,-oo), the Randič index,25 X = V2Vij(-1/2,-oo), etc. The Vij(m,n) indices which are integers The tested Vij(m,n) indices are integers when the exponents m and n form any combination of values -00, 0, 1, 2, or 3. A. Perdih, B. Perdih: Some Topological Indices Derived from the vmd"Matrix. Part 7... Acta Chim. Slov. 2003, 50, 95-114. 99 The degeneracy of Vij(m,n ) indices How much data of the tested Vij(m,n) indices is degenerated, i.e. equal to at least one more, is presented in Table 1 for ali alkanes from propane through ali octanes and in Table 2 for octanes. Highly degenerated are the indices Vij(m,-?) and Vij(m,0), and some degeneration is observed also among some other Vij(m,n) indices. The "degeneration causing" exponents are m = -?, 0, 1, and 2, as well as n = -?, -2, -1, 0, 1, and 2. The unconditionally degeneration causing exponents are presented in bold. A substantial part of degeneration is observed among isomers. Table 1. Degeneracy of Vij(m,n) indices for 38 alkanes from propane through ali octanes expressed as the number of data equal to at least one more. Empty space: no degeneration. m 3 4 24 2 ........6........ 24 4 2 1 22 2 25 8 0 < m < 1 4 i 24 ! 0 37: | 2 | j 37 | | 8 1 -1 < m < 0 4 24 -1 .......25" 24 -2 ........18 24 -? < m < -2 4........ 24 -? .......38 2 6 37 24 8 3 -? -? < n < -2 -2 -1 -Kn < 0 0 0 < n < 1 1 2 3 n Table 2. Degeneracy of Vij(m,n) indices among 18 octanes expressed as the number of data equal to at least one more. Empty space: no degeneration. m 3 4 15 2 4 15 2 1 8 2 | | 16 | | 4 | 0 < m < 1 4 15 0 18 2 18 4 -1 < m < 0 4 15 -1 10 15 -? < m < -l 4 15 18 4 17 14 4 -? -? < n < -l -1 -Kn < 0 0 0 < n < 1 1 2 3 n A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 100 Acta Chim. Slov. 2003, 50, 95-114. Correlation of Vij(m,n) indices with Vj(m,n) and W(m,n) indices The correlation coefficients between the Vij(m,n) indices tested here and the Vj(m,n) indices as well as W(m,n) indices4 (Vj(m,n) = 2*W(m,n)) are presented in Table 3 for ali alkanes from propane through octanes and in Table 4 only for octanes. In Table 3 we can see a perfect correlation when m = 0, since Vij(0,n) = Vj(0,n) = 2*W(0,n). Some high correlation coefficients are observed when n = -2. The worst correlations are observed when m < -1 and n < -2. Table 3. Correlation coefficients between W(m,n) and Vij(m,n) indices for data of alkanes from propane through octanes, in the plane of exponents m and n. m 3 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 2 0.9 0.9 0.9 0.9 0.9 © © © © © © © © © © 1 © © © © © © © © © © © © © © © 1/2 © © © G G G G G G G G © © G G 1/3 © G G G G G G G G G G G G G G 1/4 G G G © G G G G G G G G G G G 0 ..........1.......... 1 1 1 1 1 1 1 1 .........1.......... 1 1 1 1 1 -1/4 -1/3 © © © © G G G G G G G G G G G © © © © G © © © © © © © © G G -1/2 0.9 0.9 © © © © © © © © © © © © G -1 0.8 0.8 0.9 © 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 © -2 -4 -6 ......0.2...... 0.3 0.7 0.9 0.9 0.9 0.9 0.9 0.9 .....oT9..... 0.9 0.9 0.9 0.9 0.9 -0.4 -0.1 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 -0.5 0.7 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.8 0.8 0.9 0.9 -? X 0.7 0.8 0.8 0.9 0.9 0.9 0.9 0.9 .....09..... 0.8 0.8 0.8 0.8 0.9 -? -6 -4 -2 -1 -1/2 -1/3 -1/4 0 n 1/4 1/3 1/2 1 2 3 0.9: 0.9 < r < 0.99 ©: 0.99 < r < 0.999 G: 0.999 < r < 0.9999 ©: 0.9999 < r < 0.99999 ©099999 < r < 1 -0.4: -0.4 > r > -0.5 X: Division by zero Among octanes, when m = 0, Vij(0,n) = Vj(0,n) = 2*W(0,n), too, but when n = -? or 0, then the values of indices are equal for ali octanes. On the other hand, Vij(l,l) = 2Vj(l,l) - 98. Low correlation coefficients are observed in part near the diagonal m = -n, at n = 2 and m < -1, at n = -4 and -4 < m < 0, as well as at n < -4 and m < -1. A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 101 Table 4. Correlation coefficients between W(m,n) and Vij(m,n) indices for data of octanes, in the plane of exponents m and n. m 3 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 2 ......0.8...... 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 .....oT9..... 0.9 0.9 0.9 © © 1 0.9 0.9 0.9 0.9 0.9 0.9 0.8 0.7 X © © G 1 G G 1/2 0.9 0.9 0.9 © 0.9 0.8 0.3 -0.0 © © © G G G G 1/3 © © © © © 0.9 0.6 0.1 © © © G G G © 1/4 © © © © © 0.9 0.8 0.4 © © © G G G © 0 X 1 1 1 1 1 1 1 X 1 1 1 1 1 1 -1/4 -1/3 -1/2 -1 © 0.9 -0.7 G © © © © © -0.0 0.3 0.9 © G G 0.9 0.9 -0.8 G © © © © G 0.4 -0.0 0.5 © G G 0;9 0.9 -0.8 G © © © © G 0.9 0.7 -0.0 0.9 © G X -0.9 -0.8 © © © © © © © © 0.9 -0.0 0.9 l -2 -0.7 -0.7 -0.0 0.9 © © © © © © © © 0.8 0.4 0.9 -4 -6 -0.6 -0.3 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.0 0.7 0.7 -0.6 0.7 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.1 -00 ........x........ 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0 0.9 0.9 0.9 0.9 0.1 0.6 -00 -6 -4 -2 -1 -1/2 -1/3 -1/4 1/4 1/3 1/2 1 2 3 n Abbreviations as in Table 3. Table 5. The changes of values of Vij(m,n) indices of n-alkanes on increasing carbon number. Tested were the carbon numbers from C2 to C15. m m > -1 / 1 J 1 J J J J J J J J -1 =1 J J J J J J J J J -2 *l *) *) *) = ) J J J J J -4 ..........*7......... *^Bu *^Pe *^Hp *^Hp *^Bu *) *) J J -6 */ *uPe *^C8 H H H *uCn *^c8 J J -00 *o *l H H H H H H J -00 -6 -4 -2 -1 -1/2 -1/3 -1/4 0 n > 0 n / The value of Vij(m,n) increases linearly with carbon number J The value of Vij(m,n) increases progressively with carbon number l The value of Vij(m,n) decreases hyperbolically with carbon number ^Pe «-Pentane has the lowest Vij(m,n)„ value; beyond it the values are increasing * Vij(m,n)Et > Vij(m,n)Pr Vij(m,n)Et = Vij(m,n)Pr 0 The values of index are equal to zero, except for ethane — The values of index do not change with carbon number A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 102 Acta Chim. Slov. 2003, 50, 95-114. Dependence of values of Vij(m,n) indices of n-alkanes on the increase of size of the molecule Table 5 indicates whether the Vij(m,n)„ indices, i.e. the Vij(m,n) indices of «-alkanes increase or decrease with the increasing size of molecule. We can see that in majority of tested cases they increase, except in the region represented by -6 < n < -V4 and -oo < m < -4. Within and near the latter region the values of Vij(m,n) indices of ethane are greater than at least those of propane and in several cases even greater than the Vij(m,n) indices of other higher «-alkanes. There are also schematically indicated shapes of the increase or decrease. At n = -oo and m > -oo, the increase is linear with the slope of 22m+1. Among the W(m,-oo) indices4 the slope is equal to 2m. The increase of the Vij(m,0) indices can be described by a quadratic equation: Vij(m,0) = 22m *N2 + (2(m+2) - 22m)*N + 2, where N is the number of vertices in the graph of the «-alkane (i.e. its carbon number). Open remains the question whether there exists or not at each (m,n) combination, which indicates a decrease of the values of Vij(m,n)„ indices, a higher but finite carbon number where the Vij(m,n)„ index has its minimum. Changes of values of Vij(m,n) indices due to the increase of the size of molecules of other alkane isomers Table 6. The changes of values of Vij(m,n) indices of other alkanes on increasing carbon number m -2 < n < 3 + + + + + + .........."-4.......... + + +/- + + ........................+......................... ..........-6.......... + +/- - - + .........+......... .........-oo......... - - - 0 ........................+......................... n -00 -6 -4 < n < -1/3 -1/4 0 1/4c>. -B-c: -b>-c>. -Be: -b>e>. Bs: b>s>. B-s: b>-s>. -Bs: -b>s>. -B-s: -b>-s>. Sb: s>b>. S-b: s>-b>. -Sb: -s>b>. -S-b: -s>-b>. S-c: s>-c>. Se: s>e>. -S-e: -s>-e>. 0: The value of index is equal to zero NB: Does not index branching NS: Does not index the size of the molecule A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 104 Acta Chim. Slov. 2003, 50, 95-114. Table 8. Structural features, which have the lowest influence on the values of Vij(m,n) indices when the size of the molecules increases. Labels as in Table 7. m 3 -e-c -e-c -e-c ce ec ec ec ec -e-c -e-c -e-c -e-c -e-c -e-c 2 -e-c -e-c -e-c ec cb ec ec ec ¦¦TeTe......-e-c""":e:c' -e-c -e-c -e-c 1 -e-c -e-c -e-c ec cb ec ec ec -e-c -e-c -e-c -e-c -e-c -e-c 1/2 -e-c -e-c -e-c ec cb ec ec ec -e-c -e-c" -e-c b-c b-c b-c 1/3 -e-c -e-c -e-c ec cb bc ec ec -e-c -e-c b-c b-c b-c b-c 1/4 -e-c -e-c -c-e ec cb bc ec ec -e-c -e-c b-c -cb b-c b-c 0 -1/4 NB -ce ec ec -c-b -c-b -c-b -c-b NB -cb -cb -cb -cb -cb b-c -e-c -e-c -e-c ec ec ec ec ec -e-b -e-b -b-c -c-b -cb b-c -1/3 -e-c -e-c -e-c ec ec ec ec ec -e-c -e-b -b-c -c-b -cb b-c -1/2 -e-c -e-c -e-c ec ec ec ec ec -e-c -e-c -b-c -b-c -cb b-c -1 -c-e -e-c -e-c -c-e ec ec ec ec -e-c -e-c -e-c -e-b -cb -cb -2 -4 -e-b -e-b -e-c s-e -c-e -ce -c-s se -e-c -e-c -e-c -e-b -cb -cb -e-b -c-e -c-e -c-e -c-e -e-c -e-c -e-c ce ce es -e-b -cb -cb -6 -e-b -ce -c-e -c-b -c-b -c-b -e-b -e-b ec ec se -e-b -cb -cb -? ..........o.......... -cb -cb -cb -cb -cb -cb -e-b NS 0 ec ec se -cb -cb -? -6 -4 -2 -1 -1/2 -1/3 -1/4 1/4 1/3 1/2 1 | 2 3 n bc: ..>b>c b-c: ..>b>-c -b-c: ..>-b>-c cb: ..>c>b -c-b: ..>-c>-b -cb: ..>-c>b ce: ..>c>e -ce: ..>-c>e -c-e: ..>-c>-e -c-s: ..>-c>-s ec: ..>e>c -e-c: ..>-e>-c -e-b: ..>-e>-b es: ..>e>s se: ..>s>c s-e: ..>s>-e>. In Table 7 is presented the situation when the size of the molecule increases from heptane to oetane by elongation of the main chain retaining the branehed structure.There can be seen that the highest contribution has either the number of branehes (indicated by letter B) or the separation between branehes (indicated by letter S). A higher value of the structural feature contributes either to a higher inerease of the value of the index (no sign before the letter) or to a higher decrease of the value of the index (the - sign before the letter). At n = 0, only the number of branehes and the type of the branehed strueture (i.e. whether the braneh bearing carbon is tertiary or quaternary) influence the value of index. The index Vij(-?,l), on the other hand, is influenced only by the separation between branehes. The higher the separation between branehes the lower is the value of this index. A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 105 In Table 8 are presented for comparison the structural features that have a lower contribution than those presented in Table 7. In Table 8 can be seen a higher variability with exponents m and n than in Table 7. The dependence of Vij(m,n) indices on branching The increase or decrease with branching Whether the values of Vij(m,n) indices increase or decrease with branching is presented for octanes in Table 9. Table 9. Schematic presentation of the change of values of Vij(m,n) indices of octanes on increasing branching. m 3 + + + + + + + + + + + + 1- 4+ 2+ 2 1 1/2 + + + + + + + + + + + + + + + + + + + + + + + + + : 4- 2+ --- - -- -- -- -- -- - 5- 5+ - 1/3 + + + + + + 4- 6+ - - - - - - - 1/4 0 -1/4 + NB ....... + + + + + + + + + + + + + + + + 5- - - -- - --- --- --- - + + NB! - + i 1- 9- ! - -1/3 - - - + + + + + + + 1- 4+ - - - -1/2 -1 -","o"" -1+ -1+ + + + + + + + + + + + + + + + i + 1- - -- - + + - -2 2+ 2+ 7- + + + + + + + + + + 2- 1+ -4 6+ 7+ + + + + + + + + + + + + 5+ -6 6+ + + + + + + + + + + + + + 8+ -? 0 + + + + + + + + + 1/4 + 1/3 + 1/2 + 1 + 2 8- -? -6 -4 -2 -1 -1/2 -1/3 -1/4 0 n 3 + : The value of ali isomers increases on increasing branching - : The value of ali isomers decreases on increasing branching 1+ : The value of one isomer increases whereas the value of other isomers decreases on increasing branching 0: The value of index is equal to zero NB: Does not index branching Dotted lines: Intermediate region The situation in Table 9 is similar but not equal to that observed among the W(m,n) indices.4 In both cases we can observe in the plane of exponents m and n two regions A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 106 Acta Chim. Slov. 2003, 50, 95-114. where the values of indices decrease with increasing branching. They are separated from the region, where the values of indices increase with increasing branching, by intermediate regions. In the intermediate regions the values of indices either do not depend on branching or the values of indices of some isomers increase with branching whereas the values of indices of other isomers decrease with branching. The different positions of the intermediate regions among the W(m,n) indices4 compared to those among the Vij(m,n) indices seem to be responsible for low correlations presented in Table 4. The influence of structural features The comparison of values of Vij(m,n) indices when the branching increases allows some conclusions about the contribution of particular structural features. In Table 10 can be seen that the highest contribution to the value of Vij(m,n) indices due to branching has either the number of branches (indicated by letter B) or the presence of ethyl groups (indicated by the letter E) or even the separation between branches (indicated by letter S). The contribution of ethyl groups or of the separation between branches is higher than that of the number of branches mainly among the Vij(m,n) indices positioned in the intermediate region of Table 9. A higher value of the structural feature contributes either to a higher increase of the value of the index (no sign before the letter) or to a higher decrease of the value of the index (the - sign before the letter). At n = 0, only the number of branches and the type of the branched structure (i.e. whether the branch bearing carbon is tertiary or quaternary) influence the value of index. In Table 11 are presented for comparison the structural features that have a lower contribution than those presented in Table 10. The size of the molecule Only the size of molecule index the indices Vij(0,0) and Vij(0,-oo). Other Vij(m,n) indices except Vij(-oo,0) index besides the size of molecule also the influence of other structural features and in majority of cases, except in some cases when -oo < m < -4 and -oo < n < 0, cf. Table 6, the contribution of the size of molecule to the value of index is A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 107 greater than the contribution of structural features indicating branching. This fact will be kept in mind and not be mentioned again below unless considered necessary. Table 10. Two structural features having the highest contribution to the value of a Vij(m,n) index due to the increase of branching. m Be Be Be Be Be Be Be Be Be Be 3 2 1 1/2 1/3 1/4 0 -1/4 -1/3 -1/2 -1 -2 -4 -6 -? B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B B B B B B Bs B-e Bs Bs i S-e Eb Bs !~Es"i"Be""Be" Be Be Be Be Be Be Be Be Be Be Be -E-c Ec Be Be Be Be Be Be Eb -Eb -E-c -E-c B-sce Bce Bce Bce NB Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be" "B^"J Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be Be NB b- sle""B-"e B-c-e B-e B-e B-c-e B-e B-e Be Be Be Be Be Be B B B B B B B B -Eb -E-c! Be Be Be B-e B-e B-e Bs Bs Bs Bs B"eTEc":Be Be B-e B-e i Be Be B-c-e B-c-e B-c-e B-c-e 0 B-e ! -E-c -E-c Ec Be i B-s" Be B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s B-e B-e ! -Eb ! Be Bs Bs BsT-E-c Bs Bs Bs ; Sb Bs Bs Bs Bs Bs Bs Bs Bs Be Be -E-c:"Be" ' B-c" B-c Be "BeTB-c" B-c ' B-c B-c B-c B-c B-c Be Be -? -6 -4 -2 -1 -1/2 -1/3 -1/4 0 n 1/4 1/3 1/2 1 2 3 Labels: The label has four elements, e.g. b>c>e>s. In Table 11 are presented the former two in the form, e.g. Bc meaning b>c>.., whereas in Table 12 are presented the latter two, e.g. in the form es meaning ..>e>s B: b (c = e = s = 0) B-c: b>-c>. Bs: b>s>. Eb: e>b>. Sb: s>b>. 0: The value of index is equal to zero NB: Does not index branching Bce: b>c = e.. Be: b>e>. B-s: b>-s>. -Eb: e>b>. Ec: e>c>. S-e: s>-e>. B-c-e: b>-c = -e .. B-e: b>-e>. -E-c: -e>-c>. B-sce: b>-s = c = e -Es: -e>s>. Number of branches Only the number of branches indexes the index Vij(-?,0). The contribution of the number of branches to the value of other Vij(m,n) indices is in most cases the major one, Table 10. A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 108 Acta Chim. Slov. 2003, 50, 95-114. Table 11. Two structural features having the lowest contribution to the value of a Vij(m,n) index due to the increase of branching. m 3 2 1 1/2 v3 1/4 0 -1/4 -1/3 -1/2 -1 -2 -4 -6 -? c=e c=e -s -s -s NB s s s s s s s 0 ec ec ec ec ec ec ec ec "c:s'"c-s"c-s'"c-s' c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s "cs~"~-c7j c-s c-s ec ec ec '• -e-c -e-c -e-c i -cb '• c-s c-s ""^""sc^sci ^c-s'"c-š'"c-š] -cs |-e-c !-cb i c-/ c-s c-s ~~c-s~T-~s~~~c~~-r c-s c-s c-s c-s 1 s-b.....-cs i c-s | b-s sb j c-s c-s s-b i c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s c-s \ NB c-s c-s c-s j c-s c-s c-s j c-s c-s c-s \ c-s c-s c-s c-s c-s c-s -cs s-b c-s c-s c-s c-s -cs -cs -cs -cs c-s c-s c-s c-s -cs -cs ! b-s ! c-s c-s c-s -cs -cs -cs : c-s c-s c-s -cs |~bs" ^sclec"" ""ec""""ec""""ec"! s-c s-c -cs -csic* c-s '"blTb"-!" -c-e -c-e -c-e j -c-e -c-e -c-e j -c-e -c-e -c-e i .e-c -e-c -e-c -e-c 1 s-b J c-s -bs 1 -s-c ce ce ce j -e-c -e-c j c-s -s-e -s-e ce ce ce ! -e-c -e-c '• c-s ":e:s.....~-e"š" ce ce ce "c=_e i c-s -? -6 -4 -2 -1 -1/2 -1/3 -1/4 0 1/4 1/3 1/2 1 2 3 n bs: ..>b>s b-s: ..>b>-s -bs: ..>-b>s -cb: ..>-c>b ce: ..>c>e c-e: ..>c>-e -c-e: ..>-c>-e -cs: ..>-c>s c-s: ..>c>-s ec: ..>e>c -e-c: ..>-e>-c -e-s: ..>-e>-s s: ...>s -s: ...>-s sb: ..>s>b s-b: ..>s>-b s-c: ..>s>-c -se: ..>-s>c -s-c: ..>-s>-c -se: ..>-s>e -s-e: ..>-s>-e 0: The value of index is equal to zero NB: Does not index branching The type of the branehed strueture The size of molecule, the number of branehes as well as the type of the branehed strueture, i.e. whether the braneh bearing carbon is tertiarv or quaternary index the Vij(m,0) indices, which are not mentioned above. They indicate that the strueture having a quaternary carbon is more branehed than that having two tertiarv carbons. The index Vij(2,0) indicates that the strueture having three tertiarv carbons is equally branehed as that having one quaternary carbon. The type of branehes The label E in Table 10 indicates that at some combinations of exponents m and n in the Vij(m,n) indices the exchange of a methyl group for an ethyl group in the A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 109 structure of an octane contributes to the value of index more than any other structural feature, whereas in several cases (label e) it is the second greatest contribution. Position of branches Position of branches contributes to the value of a Vij(m,n) index in most cases less than the number of branches and the type of branches. Separation between branches The separation between branches contributes to the value of the indices Vij(3,l) and Vij(-4,2) more than any other structural feature contributing to branching. It has the second greatest contribution to the values of several indices when m < -1 and -1 < n < 2. Correlation of physicochemical properties with Vij(m,n) indices The values of tested Vij(m,n) indices were correlated with values of 23 or 24 physicochemical properties, as applicable, assuming a linear relationship between them. The results are presented in Fig. 1 for data of propane through octanes and in Fig. 2 for data of octanes. The data of 2,2,3,3-tetramethyl butane are lacking at MON and logVP. In these cases, 2,2,3,3-tetramethyl butane is not considered in correlation. The pattern in Fig. 1 is quite different from that in Fig. 2. As presented by Fig. 1, molecular weight correlates perfectly with the index Vij(0,-oo) which indicates only the size of molecule and increases linearly with carbon number. Eleven other physicochemical properties correlate best with other Vij(m,-oo) indices increasing linearly with carbon number with the slope of 22m+1. Four of them correlate best with the index Vij(-72,-oo) = 2%. Additional five physicochemical properties correlate best with Vij(m,-6) indices that do not deviate much from the linear increase with carbon number. Most of mentioned indices have a negative value of exponent m. The best correlations with Vij(m,n) indices having a positive value of the exponent n have only the MON and ra. The best correlations are observed at Mw (r = 1) > MR (r = 0.998), Tc2/Pc (r = 0.998) > AHv (r = 0.997) > BP (r = 0.994), Tc/Pc (r = 0.994) > AHPg (r = 0.991), logVP (r = -0.990). Most of them are slightly higher than with the W(m,n) indices.4 A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7... 110 Acta Chim. Slov. 2003, 50, 95-114. The pattern of Fig. 2 is different from that of Fig. 1. Evidently, the pattern of Fig. 1 is governed mainly by the dependence of the values of physicochemical properties on the size of the molecule. The pattern in Fig. 2, on the other hand, is governed only by branching since the influence of the size of the molecule is excluded. Surprising is the emptiness of the lower central part of the figure. The best correlations are observed at Tc/Pc (r = -0.998), co (r = -0.996), BP/Tc (r = -0.995), MON (r = -0.987), Tc2/Pc (r = 0.984), C (r = 0.969), S (r = -0.961), and Pc (r = 0.950). m 3 2 nD 1 v2 d MON v3 AH° v4 co 0 M -V4 P V -v3 BP T2/P -V2 a VP Te -1 Zc de B -2 -4 -6 A -00 0 -00 -6 -4 -2 -1 -V2 -V3 -V4 v4 v3 v2 1 2 3 n Figure 1. The positions of 23 physicochemical properties determined by the highest correlation coefficient r (data in parentheses, see below) for data of alkanes from propane through oetanes, in the plane of exponents m and n. Single standing: nD (0.910), d (0.926), MON (-0.756), AHf°g (0.991), co (0.948), BP (0.994), Tc2/Pc (0.998), logVP (-0.990), Te (0.986), Zc (-0.732), de (0.802), B (0.981), A (0.651) M: Mw (1), MR (0.998) P: Tc/Pc (0.994), Pc (-0.957) V: Ve (0.988), Vm (0.979) a: ac (0.936), BP/Tc (0.963), C (-0.959), AHv (0.997) The comparison of Fig. 2 here with Fig. 3 in ref4 shows that there are some similarities but also several differences in positions of best correlations between the two groups of indices and physicochemical properties. The best correlation coefficients are higher for Vij(m,n) indices and the following physicochemical properties of oetanes: Zc, A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… Acta Chim. Slov. 2003, 50, 95-114. 111 Te, BP, AHv, BP/Tc, d, Vm, and nD; they are higher for W(m,n) indices and the following physicochemical properties of oetanes: A, B, MR, and AHf°g. For other physicochemical properties the correlations are very similar at both groups of indices. m 3 2 1 v2 v3 v4 o -V4 -V3 -V2 -1 -2 -4 -6 -00 dVn B/T C CD ......BP vpa S T/P Pc .oo .6 -4 -2 -1 -V2 -V3 -V4 0 V4 V3 V2 1 2 3 n Figure 2. The positions of 24 physicochemical properties determined by the highest correlation coefficient r (data in parentheses, see below) for data of oetanes, in the plane of exponents m and n. Single standing: Zc (0.694), Te (-0.830), MR (0.882), Pc (0.950), ac (0.740), AHf°g (-0.904), R2 (0.907), MON (-0.987), C (0.969), q (-0.996), BP (0.885), AHv (0.936), S (-0.961), B (-0.697) B/T: BP/Tc (-0.995) T/P: Tc/Pc (-0.998) T2/P: Tc2/Pc (0.984) dVn: d (0.915), Vm (-0.918), nD (0.899) vpa: logVP (-0.738), A (0.743) #: Ve (0.844), de (-0.831) Dotted lines: Intermediate domains from Table 9. The Vij(m,n) indices that might be good branehing indices In Fig. 3 are presented some interesting characteristics of the Vij(m,n) indices in the plane of exponents m and n. The index Vij(-oo,0) indexes only the number of branehes, Vij(-oo,0) = 2JLNp-\) = 2^(\+Nbr). It is a simple, primitive and degenerated, but a true branehing index presenting the most important information about branehing - the number of branehes (the number of vertices of degree one). A. Perdih, B. Perdih: Some Topological Indices Derived from the vmdn Matrix. Part 7… 112 Acta Chim. Slov. 2003, 50, 95-114. The indices Vij(m,0), n * -oo and n * 0, index the size of the molecule, the number of branches and in addition they indicate that a quaternary structure is more branched than a tertiary one. They are degenerated branching indices, less simple than Vij(-oo,0). m 3 ^ BI 2 =(M) 1 ^ 1/2 ^ 1/3 ^ 1/4 ^ 0 NB (H) (RW) -1/4 ^ 5/ BI BI BI BI -1/3 ^ BI BI BI BI -1/2 =(x) BI BI BI BI -1 ^ BI BI BI BI -2 ^ # -4 ^ -6 ^ -00 0 -00 -6 -4 -2 -1 -1/2 -1/3 -1/4 r<<7 3f=2g t 2M > 3M > 3E > 24M > 23M > 22M > 33M > 223M S7;8: n- > 2M > 3M > 3E > 22M > 33M > 23M > 24M > 223M Oct < 3E6 < 4M7 < 3M7 < 2M7 < 3E3M5 < 33M6 < 22M6 < 3E2M5 < 34M6 < 23M6 < 24M6 < 25M6 < 233M5 < 223M5 < 234M5 < 224M5 < 2233M4 Oct < 2M7 < 3M7 < 4M7 < 3E6 < 22M6 < 33M6 < 3E3M5 < 23M6 < 3E2M5 < 34M6 < 24M6 < 25M6 < 223M5 < 233M5 < 224M5 < 234M5 < 2233M4 Oct < 2M7 < 3M7 < 4M7 < 3E6 < 22M6 < 33M6 < 3E3M5 < 23M6 < 3E2M5 < 34M6 < 24M6 < 25M6 < 223M5 < 233M5 < 234M5 < 224M5 < 2233M4 n- < \t < 2q (sep. = 0) < 2t (sep. = 1) < 2t (sep. = 2) < 2t (sep. = 3) < 2q\t (sep. = 1) < 2q\t=3t (sep. = 2) < 4q (sep. = 1). Thus, the influence is B > s, cf. Table 10 and 11; the position of branches is not important Only the number of branches influences the value of this index 0: The value of index is equal to zero NB: Does not index branching 0 ^ BI BI t