Acta Chim. Slov. 2002, 49, 467-482. 467 SOME TOPOLOGICAL INDICES DERIVED FROM THE vmdn MATRIX. PART 4. THE LARGEST EIGENVALUES OF THE "MEAN DEGREE OF VERTICES " MATRICES AS TOPOLOGICAL INDICES OF THE BIM-TYPE Anton Perdih,* Branislav Perdih Mala vas 12, SI-1000 Ljubljana, Slovenia Received 12-11-2001 Abstract The largest eigenvalues of the "mean degree of vertices" matrices form a group of the BIM type indices. They are good indices for indexing the influence of the size of molecule, but less good as branching indices. True but degenerated branching indices are the indices Aa.(0), Ga.(0), Ln(0), RA^(0), RG^(0), RLn(0), and RZg^(0). Regular sequences of isomers on increasing branching have only the indices RA^(-1), RG(-1; -2), and RLn(-1). Introduction One of the most useful topological indices besides the Wiener1 index W is the Randić index x.2,3 In view of the formalism of the vmdn type of matrix,4 the Wiener index W is derived by summation of the elements of the matrix having the nondiagonal elements gij = (vixvj )0xdij 1, whereas the Randić index x is derived by summation of the elements of the matrix having the nondiagonal elements gij = (vixvj )-1/2xdij -00, where v is the degree of vertex and dij is the shortest distance from vertex i to vertex j . Randić index x can be considered also as a "reciprocal geometric mean of the degree of adjacent vertices" type of summation-derived indices. Looking at the index x from this point of view, there arises the question whether there exist also some other good indices of the "mean degree of vertices" type having dij raised to other values of exponent. In a previous paper5 we considered the summation-derived "mean degree of vertices" type indices. In present paper we consider the indices, which are the largest eigenvalues of the same matrices. Data and methods The structures of alkanes are presented in shorthand, e.g. Hp is n-heptane, Oct is n-octane, 223M5 is 2,2,3-trimethylpentane, 3Et2M5 is 3-ethyl-2-methylpentane, etc. The other terms are explained on 2,2-, 2,3- and 2,5-dimethyl hexane (22M6, 23M6 and 25M6) as examples. The two branches (the number of branches, Nbr = 2) in 22M6 are A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 468 Acta Chim. Slov. 2002, 49, 467-482. positioned on a quaternary carbon (2q) placed on the periphery (per) of the molecule. The two branches in 23M6 and 25M6 are positioned on tertiary carbons (2t). Regarding the separation between branches (s), in 23M6 the branches are adjacent (adj) and those 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. 25M6 is symmetric, whereas 22M6 and 23M6 are not. Since we only consider alkanes, we use the words "vertex" and "carbon" as synonyms that reflect different contexts. The use of "vertex" implies the graph-theoretical consideration, whereas the use of "carbon" implies the chemical consideration of the same thing. Their number is given as NC or Nv, respectively. The susceptibility for the increase in carbon number, Si , i+1, and the susceptibility for branching, Sn,i, is defined as the normalised difference of the indices' or properties' values.4,6 As criteria regarding the degree of branching is used the intuition7 as well as the Methane based definition and the n-Alkane based definition8 of branching. The physicochemical properties The data for the boiling point (BP), density (d), the critical data Tc, Pc, Vc, Zc, oc, and dc, 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 refractive index (nD) were taken from the CRC Handbook9 or from Lange's Handbook10. 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 refraction (MR) were calculated from data presented in the handbooks. The data for Motor Octane Numbers (MON) was taken from Pogliani11 and Ren,12 those for vapour pressure (logVP) from Goll and Jurs,13 and those for the entropy (S) and quadratic mean radius (R2) from Ren.12 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 469 Results and discussion The "mean degree of vertices" matrix and its largest eigenvalue as a topological index The vmdn matrix4 is a matrix having the main diagonal elements, gii = 0, and all the others, gij (irj) = vj mxdi j n, where vj is the degree of vertex j (in alkanes it is the number of the C-C bonds which the carbon in question is involved in) and dij is the shortest distance from vertex i to vertex j (in alkanes it is the lowest number of bonds between the carbons in question). A more general type of the vmdn matrix is the matrix having the elements gij (irj) = vi axvj bxdij c. A representative of the indices, summation-derived from this matrix, is one of the most useful indices, the Randić2 index x, having the exponents a = -1/2, b = -1/2, c = -oo, whereas its zero order variant, the index 0x, has the exponents a = -1/2, b = 0, c = -oo. The Randić2 index x is thus a "reciprocal geometric mean of the degree of adjacent vertices" index,4 derived from the matrix having the non-diagonal elements gij (&j) = (viXv j )-1/2xdij-". Due to the great success of this index, it seemed plausible to test also other indices derived by using other types of the "mean of the degree of vertices", denoted also as "mean(v i ,vj)", where gij (irj) = ("mean(v i ,vj)"^ij c. On the one hand, the indices were derived using as the "mean(v i ,vj)": - The arithmetic mean (average) of the degrees of vertices, (vi+vj)/2, the indices being labelled as AE or AE(c), - The geometric mean, (vixvj)1/2, the indices being labelled as Gs or Gs(c), - The logarithmic mean (vi -vj )/(lnvi -lnvj), the indices being labelled as Lns or Lns(c). For comparison, the Zagreb14 product (vixvj)2 was included as a "mean(v i ,vj)" and the indices derived using it were labelled as ZgE or ZgE(c). The indices were derived also using the reciprocal values of "mean(v i ,vj)"; so they were labelled as RAS, RGS, RLns, RZgs or RAs(c), RGs(c), RLns(c), RZgs(c), respectively. On the other hand, the tested exponents c in dij c were: -oo, -16, -12, -10, -8, -6, -4, -2, -1, 0, 1, 2, and 3. The shorter labels, e.g. AE, were used when speaking about that group of indices as a whole, whereas the forms as e.g. As(c), were used when speaking about that index at the value of exponent c specified in parentheses. The general label for all these indices was XYs(c). A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 470 Acta Chim. Slov. 2002, 49, 467-482. In present paper we consider the indices A?(c), G?(c), Ln?(c), Zg?(c), RA?(c), RG?(c), RLn?(c), and RZg?(c), which are the largest eigenvalues of the same matrices. A common label for these indices is XY?(c). Characteristics of the largest eigenvalues as "mean degree of vertices" indices The values of the "mean degree of vertices" indices of methane are equal to zero and those of ethane are equal to 1. Integers, i.e. all equal to 1, are the values of RG?(-?). The degeneration of the largest eigenvalues as the "mean degree of vertices" indices The number of degenerated data, i.e. of data having at least one equal counterpart among all tested XY?(0) indices of alkanes from propane through octanes is equal to 24, whereas among octanes it is equal to 15. The index RG?(-?) is totally degenerated. Correlation between the largest eigenvalues and the summation-derived "mean degree of vertices" indices derived from the same matrices The correlation coefficient between the data of the largest eigenvalues and the data of the summation-derived "mean degree of vertices" indices derived from the same matrices of alkanes from propane through octanes and assuming the linear relationship, is presented in Table 1, whereas for octanes it is presented in Table 2. Table 1. Correlation between the largest eigenvalues and of summation-derived "mean degree of vertices" indices derived from the same matrices of alkanes from propane through octanes. Index c Ax(c) Gx(c) Lnx(c) Zgx(c) RAx(c) RGx(c) RLnx(c) RZgx(c) 3 0.9931 0.9934 0.9933 0.9839 0.9898 0.9895 0.9896 0.9562 2 0.9918 0.9920 0.9919 0.9695 0.9912 0.9905 0.9908 0.9600 1 0.9914 0.9909 0.9911 0.9515 0.9914 0.9915 0.9915 0.9866 0 0.9857 0.9875 0.9870 0.9316 0.9806 0.9824 0.9819 0.9737 -1 0.9643 0.9722 0.9697 0.9382 0.9654 0.9575 0.9607 0.8875 -2 0.9409 0.9504 0.9472 0.9499 0.9682 0.8973 0.9245 0.5341 -4 0.9149 0.9214 0.9191 0.9596 0.5489 0.6892 0.9563 0.2227 -6 0.9065 0.9116 0.9098 0.9617 0.3290 0.5595 0.3510 0.5056 -8 0.9043 0.9091 0.9073 0.9621 0.3056 0.5097 0.2054 0.5380 -10 0.9037 0.9084 0.9067 0.9622 0.3016 0.4911 0.1799 0.5442 -12 0.9036 0.9082 0.9066 0.9622 0.3008 0.4839 0.1742 0.5457 -16 0.9035 0.9082 0.9065 0.9623 0.3005 0.4797 0.1725 0.5461 -00 0.9035 0.9082 0.9065 0.9623 0.3005 0.1724 0.5461 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 471 Among the indices of the groups A?(c), G?(c), Ln?(c), RA?(c), RG?(c), and RLn?(c), the correlations presented in Table 1 are quite good when c ? 0 and at increasingly negative values of exponent c they decrease to a limiting value. Among the indices of the groups Zg?(c) and RZg?(c) the correlations are not as good. The reason for this difference may be in the fact that the overall dimension in the degree of vertex is in the former group v1 or v-1, whereas in the latter group it is v4 or v-4. Some aspects of the influence of the exponent in vm were described elsewhere.4 If only octanes are considered, several correlation coefficients are higher, indicating that several largest eigenvalues of tested matrices contain the information very similar to that of the summation-derived indices of this type. Table 2. Correlation between the largest eigenvalues and of summation-derived "mean degree of vertices" indices derived from the same matrices of octanes. Index c Ax(c) Gx(c) Lnx(c) Zgx(c) RAx(c) RGx(c) RLnx(c) RZgx(c) 3 0.9989 0.9989 0.9989 0.9701 0.9976 0.9972 0.9974 0.9415 2 0.9996 0.9996 0.9996 0.9011 0.9973 0.9970 0.9971 0.4633 1 0.9989 0.9996 0.9995 0.5688 0.9734 0.9899 0.9872 0.9639 0 0.9961 0.9927 0.9897 0.9995 0.9998 0.9997 0.9940 -1 0.9976 0.9990 0.9995 0.9854 0.9996 0.9998 0.9998 0.9969 -2 0.9860 0.9924 0.9887 0.9869 0.9946 0.9991 0.9986 0.9937 -4 0.9603 0.9706 0.9632 0.9886 0.9921 -0.8854 0.4522 0.6517 -6 0.9498 0.9616 0.9525 0.9891 0.9867 -0.9575 0.9957 0.7682 -8 0.9470 0.9592 0.9497 0.9892 0.9833 -0.9653 0.9960 0.7678 -10 0.9464 0.9587 0.9490 0.9892 0.9823 -0.9684 0.9956 0.7674 -12 0.9462 0.9586 0.9488 0.9892 0.9821 -0.9699 0.9955 0.7673 -16 0.9461 0.9585 0.9488 0.9892 0.9820 -0.9708 0.9955 0.7673 -00 0.9461 0.9585 0.9488 0.9892 0.9820 0.9955 0.7673 The increase of values of the largest eigenvalues of the "mean degree of vertices" matrices of n-alkanes due to the increase of size of the molecule How the values of the largest eigenvalues of the "mean degree of vertices" matrices of n-alkanes from propane through octanes increase with increasing carbon number is presented in Table 3 for all tested indices. Whereas the tested XYE(c) indices5 have at c = -qo a linear increase with carbon number, and at -oo < c < -6 the increase is apparently but not truly linear, the tested XYx(c) indices have a declining increase of values, with RZgx(c) which decreases with A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 472 Acta Chim. Slov. 2002, 49, 467-482. carbon number, as an exception. At c = 0, the increase of the tested XYs(c) indices is quadratic, whereas most XYx(c) indices are increasing either slightly more or slightly less than linearly. These differences are the main cause for lower correlation coefficients, Table 1, when c < 0. At c > 0, the increase can be described by a higher polynomial. Among the XYx(c) indices it is in several instances of lower degree than among the XYs(c) indices. Table 3. Schematic presentation of the shape of the increase of values of the "mean degree of vertices" XYx(c) indices of n-alkanes from C3-C8 as well as of the ________degree of the simplest polynomial having r2 > 0.9995.__________ _____________________c_________________ Index___________________-oo Oct, 2M6 -> 2M7, 3M6 -> 3M7, 3Et5 -> 3Et6, 24M5 -> 24M6, 23M5 -> 23M6, 22M5 -> 22M6, 33M5 -> 33M6, and 223M4 -> 223M5. In these transitions the main chain of the alkane is elongated by insertion of a methylene group, so that the branched structure is preserved as much as possible although it is shifted to the periphery of the molecule. On increase of the size of molecule increase also the values of most of largest eigenvalues of the "mean degree of vertices" matrices tested here, Table 4. The exceptions are the indices RG^(c) when -oo < c < -4 and RZgx(c) when c < -2, where some values of indices increase and the others A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 473 decrease. At RZg?(-2) and RZg?(-1) all values decrease on increasing size of the molecule. Table 4. The increase of values of the largest eigenvalues of the "mean degree of vertices" matrices on increase of the size of molecule. c Index -00 -oo< c<-4 -2e S>e S>e S>e S>e S>e a1 s s s G^(c) S>e S>e S>e S>e S>e S>e a S S>e S Lnx(c) S>e S>e S>e S>e S>e S>e a S>e S S Zgx(c) S>e S>e S>e S>e S>e S a2 S S S RAx(c) s s s S S>e s a1 S>e S S RGx(c) = BI s s s s a1 S>e S>e S RLnx(c) S S S S S s a1 S>e S>e S RZgx(c) S S S S l s a1 S>e S>e S a1: 0 > 1t > 2t > 2q > 2q1t = : no change a: 0 < 1t < 2t < 2q < 2q1t a2: 2q > 0 > 1t > 2t > 2q1t BI: 0 > 2M > 3M > 3E > 24M > 23M > 22M > 33M > 223M Symbols for structural features: e: ethyl vs. methyl l: position of branches (ctr) s: separation between branches Uppercase symbols: That structural feature has the highest influence Lowercase symbols: That structural feature has the highest influence (next to the number of branches, if not shown explicitly otherwise) In Table 5 is presented, which structural feature has the highest influence on the increase of values of the largest eigenvalues of the "mean degree of vertices" matrices on increase of the size of molecule. In cases marked "a", "a1" or "BI" the sequence of A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 474 Acta Chim. Slov. 2002, 49, 467-482. isomers seems regular as it would be expected for a good branching index. The former situation is observed when the exponent c = 0 at the indices G^ and Lnx, the next one at the indices Ax, RAX, RGX, RLnx, and RZgx. In these cases only the number of branches and the type of the branched structure influence the values of the indices. The last situation is observed at the index RG^(c) when -oo < c < -6. There are also given some qualitative conclusions which structural feature has a higher influence on the change of the index value. When c > 0, then at the index Ax the highest influence has the number of branches followed by the separation between them. Similar situation is among the indices RAX when c = -1 or -oo < c < -6, RGx when -oo < c < 0, at RLnx(-1) and RZgx(-1). At RZgx(-2), the number of branches has the highest influence, followed by the position of branches. The dependence of the largest eigenvalues of the "mean degree of vertices" matrices on branching The increase or decrease with branching Whether the largest eigenvalues of the "mean degree of vertices" matrices increase or decrease with branching is presented in Table 6. Table 6. The values of the largest eigenvalues of the "mean degree of vertices" matrices increase or decrease with branching c Index -00 -16 < c <-6 -4 -2 -1 0 1 2 3 Ax(c) + + + + + + - - - Gx(c), Lnx(c) + + + + + - - - - Zgx(c) + + + + + + ± ± - RAx(c) - - - + + + ± - - RGx(c) = + + + + + ± - - RLnx(c) - - ± + + + ± - - RZgx(c) - - ± + + + + ± - +: The values increase on increasing branching -: The values decrease on increasing branching ±: Some values increase whereas the other ones decrease when the branching increases A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 475 Most of them decrease when c > 0. The indices RA?(c), RLn?(c), and RZg?(c) decrease also when c < -2 or -4, respectively. The values of the indices A?(c), G?(c), Ln?(c), Zg?(c), and RG?(c) increase with branching when c < 0. The influence of structural features Which structural feature influences the value of the tested "mean degree of vertices" indices most strongly is presented in Table 7. In majority of cases it is the number of branches (a, a1, BI and lowercase symbols in Table 7) followed by either the type of branches (ethyl vs. methyl) or the separation between branches. The position of branches seems to have in most cases a lower influence than the other structural features. Table 7. Which structural feature influences the values of the "mean degree of vertices" indices the most strongly. C Index -00 -162 Ax(c) s s s e e a e e Gx(c), Lnx(c) s s s e e a1 e e Zg^(c) s s s s s a E>l e RAx(c) s s e e BI a s s RGx(c) 0 s s BI BI a e e RLnx(c) e e E>l e BI a e e RZgx(c) S>l S>l s s l a s e a: 0 < 1t < 2t < 2q < 3t < 2q1t < 4q a1: 0 > 1t > 2t > 2q > 3t > 2q1t > 4q BI: Oct < 2M7 < 3M7 < 4M7 < 3Et6 < 25M6 < 24M6 < 23M6 < 34M6 < 3Et2M5 < 22M6 < 33M6 < 3Et3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4 Symbols for structural features: e: ethyl vs. methyl l: position of branches (ctr) s: separation between branches Uppercase symbols: That structural feature has the highest influence Lowercase symbols: That structural feature has the highest influence (next to the number of branches, if not shown explicitly otherwise) The largest eigenvalues of the "mean degree of vertices" matrices that might be good branching indices Taking into account the information presented in Table 5 (RG?(-16 ? c ? -8)) and Table 7 (RG?(-2 and -1) can be concluded that none of tested largest eigenvalues of the "mean degree of vertices" matrices has an ideal full sequence of isomers on increasing A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 476 Acta Chim. Slov. 2002, 49, 467-482. the size of molecule and on increase of branching, so none is a perspective candidate for an ideal branching index of BIM8 type. On the other hand, the indices RA^(-1), RGx(-1), RGx(-2), and RLnx(-1) have an ideal full sequence of isomers on increase of branching making them candidates for a good branching index of BIA8 type. The indices Ax(0), Gx(0), Lnx(0), Zgx(0), RAx(0), RGx(0), RLnx(0), and RZgx(0) have an ideal sequence of isomers presenting the information that the number of branches has the highest contribution and the quaternary structures contribute more than the tertiary ones. No other information is contained in them. Correlation of physicochemical properties with the largest eigenvalues of the "mean degree of vertices" matrices The correlation of the largest eigenvalues of the "mean degree of vertices" matrices with the tested physicochemical properties assuming the linear relationship was derived using data of all alkanes from ethane through octanes, as well as only for octanes. The patterns are similar to those observed at the indices XYE.5 The differences are as follows. Data from ethane through octanes: At c < -4 are the correlations lower among the indices Zgx, RAX, RLnx, and RZgx. At RAX, RLnx, and RZgx some curves cross the abscissa at c around or below -4. When only data of octanes is correlated, we observe two patterns. Abscissa is crossed once at the indices Ax, Gx, Lnx, RGX, and Zg*. at 0 < c < 1, -1 < c < 0, -1 < c < 0, 0 < c < 1, 0 < c < 3, respectively. Abscissa is crossed twice at the indices RAX, RLnx, and RZgx at -4 < c < -2 and 0 < c < 2, at -6 < c < -2 and 0 < c < 2, as well as at -6 < c < -2 and 1 < c < 2, respectively. The values of correlation coefficients differ from one physicochemical property to another, as well as from one tested index to the other one. Usually we are interested in the best correlation coefficients. This type of data is collected in Table 8-10. In Table 8 are presented the values of the exponent c, which gives rise to the best correlation of the particular index and physicochemical property, when the data of alkanes from ethane through octanes is considered. The "best" exponents are in most cases 0 < c < 2, in other cases they are mostly close to these values. A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 477 Table 8: The value of exponent c where the tested physicochemical property has the best correlation coefficient (rbest) with the particular tested largest eigenvalue of the "mean degree of vertices" matrices for alkanes, XYx(c), from ethane through octanes. Physicochem. c property Ax(c) Gx(c) Lnx(c) Zgx(c) RAx(c) RGx(c) RLnx(c) RZgx(c) BP 0 0 0 1 1 1 -4 2 AHv 1 0 0 1 1 1 1 2 AHf°g 0 -1 0 1 0 0 0 1 Tc 0 0 0 1 1 1 -4 2 Pc 0 0 0 1 1 1 1 2 Vc 0 0 0 1 1 1 1 2 dc -2 -4 -2 0 -2 -1 -2 1 oc 1 1 1 1 1 1 1 2 Zc 1 1 1 2 2 2 2 2 BP/Tc 1 1 1 2 2 2 2 2 Tc/Pc 0 0 0 1 1 1 1 2 Tc2/Pc 0 0 0 1 1 1 1 2 CO 1 1 1 2 2 2 2 3 d 0 -1 -1 0 0 0 0 1 Vm 0 0 0 1 1 1 1 2 A 3 2 2 3 -4 3 -6 3 B 0 0 0 1 1 1 1 2 C 1 1 1 2 1 1 1 2 nD -1 -1 -1 0 -2 0 0 1 MR 0 0 0 1 1 1 1 2 MON 3 3 3 3 3 3 3 3 logVP 0 0 0 1 1 1 1 2 Mw 0 0 0 1 1 1 1 2 In Table 9 are presented the results for the case when only data of octanes is considered. The "best" exponents differ more widely than in Table 8 and it could be reasonably expected that some correlations might be still better at c > 3. In Table 10 are collected the cases of "best" correlations for data of all alkanes from ethane through octanes as well as when only octanes are considered. In bold are presented those physicochemical properties that are considered to be the best references for branching.6 When all tested alkanes are considered, in 18 out of 23 cases | r | > 0.9 and in five cases even | r | > 0.99. When only data of octanes is considered, then I r | > 0.9 is observed in 10 out of 24 cases and | r | > 0.99 in two cases. Consequently, several of the tested indices, which are the largest eigenvalues of the "mean degree of vertices" matrices, are good indices of the influence of size of the molecules and only A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 478 Acta Chim. Slov. 2002, 49, 467-482. few of them also indicate well branching. Branching is indicated well only in connection with the physicochemical properties that are considered to be the best references for branching, i.e. with Tc/Pc, co, and BP/Tc, followed by Tc2/Pc, MON, C, S, Pc, AHv, and R2. This is the case with the indices Gx(-1), RLnx(-2), Lnx(-2), Gx(1), etc, respectively. The influence of branching on the values of AHf°g is not indexed as well by these indices. Table 9. The value of exponent c where the tested physicochemical property has the best correlation coefficient (rbest) with the particular tested largest eigenvalue _________of the "mean degree of vertices" matrices for alkanes, XYx(c), among octanes. Physicochem. __________________________c_______________________________ property_______A*,(c) Gx(c) Lnx(c) Zgx(c) RAx(c) RG^(c) RLnx(c) RZg^(c) BP 0 0 0 1 -4 -4 -6 1 AHv 0 0 0 0 -4 -16 -6 -1 AHf°g 0 0 0 1 -4 -16 -6 -2 Tc 1 -oo -4 1 1 1 -4 2 Pc 1 -1 -1 3 1 1 1 2 Vc 3 -1 3 3 1 1 1 3 dc 2 -1 3 3 1 1 1 2 oc 3 3 3 3 3 3 3 3 Zc 0 0 0 0 -oo 0 -oo -1 BP/Tc -1 -2 -2 -1 -2 2 -2 3 Tc/Pc 1 -1 -1 -2 2 2 2 3 Tc2/Pc 2 1 1 0 3 0 0 0 CO -1 0 -1 -1 -1 -2 -2 -1 d -1 -oo -2 -oo -2 -2 -2 2 Vm -1 -oo -4 -oo -2 -2 -2 2 A 0 3 3 0 3 3 0 1 B 0 0 0 1 -4 0 -4 1 C 0 0 0 0 -1 -6 -1 -1 nD -1 -oo 0 -oo -2 -2 -2 -2 MR 1 -1 -1 2 1 1 1 2 MON 3 3 3 0 3 3 3 3 logVP 0 0 0 0 0 0 -6 1 S 0 0 0 0 -1 -6 -1 -2 R2 3 3 3 0 3 3 3 1 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 479 Considering 23 tested physicochemical properties, for indexing the influence of the size of molecules are the best indices Gx (15 cases), Lnx (5 cases), and Ax (3 cases), followed by Zgx, RAX, RGX, and RLnx (1 case). For indexing of branching are the best ones the indices from the group Zgx (8 cases), followed by RLnx (6 cases), Gx (5 cases), RAx and RZgx (2 cases), and Lnx (1 case). For indexing the size of molecules are the best exponents c = 0 (15 cases), c = 1 (8 cases), c = -4 (2 cases), c = 3 and -1 (one case each). For indexing of branching are the best exponents c = 3 (5 cases), c = -6 (4 cases), c = -oo, -1 or 1 (3 cases each), c = -2 or 2 (2 cases each), and c = -4 or 0 (one case each). Table 10. The cases of the best correlation coefficients (rbest) between tested physicochemical properties and the largest eigenvalues of the "mean degree of vertices" matrices, XYx(c), among alkanes from ethane through octanes as well as only among octanes. Physicochem. rbest C2-C8 index c C8 property rbest index c BP 0.989 Gx 0 0.824 RLnx -6 AHv 0.986 Gx 0 0.935 RLnx -6 AHf°g 0.989 Ax 0 -0.860 RLnx -6 Tc 0.981 Lnx, Gx 0 0.831 RLnx -4 Pc -0.950 Gx 0 -0.955 RZgx 2 Vc 0.991 Gx 0 0.841 Zgx 3 dc 0.899 Gx -4 -0.824 Zgx 3 oc 0.951 Ax 1 0.740 Gx 3 Zc -0.744 Gx 1 0.686 Zgx 0 BP/Tc 0.965 Lnx 1 -0.989 Lnx -2 Tc/Pc 0.993 Gx, RGx, RLnx 0, 1, 1 -0.996 Gx -1 Tc2/Pc 0.997 Gx 0 0.985 Gx 1 Q 0.971 Gx 1 -0.993 RLnx -2 d 0.929 Gx -1 0.884 Zgx -oo Vm 0.980 Gx 0 -0.890 Zgx -oo A 0.716 RAx -4 -0.775 RZgx 1 B 0.979 Gx 0 -0.723 Zgx 1 C -0.954 Ax, Lnx 1 0.968 RAx -1 nD 0.886 Zgx 0 0.860 Zgx -oo MR 0.998 Lnx 0 0.891 Zgx 2 MON -0.804 Gx 3 -0.984 Gx 3 logVP -0.986 Gx 0 -0.727 RLNx -6 S -0.961 RAx -1 R2 0.917 Gx 3 Mw 0.999 Lnx 0 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 480 Acta Chim. Slov. 2002, 49, 467-482. Correlation between the largest eigenvalues of tested "mean degree of vertices" matrices In Tables 3-10 can be seen that the largest eigenvalues of tested "mean degree of vertices" matrices share many similar characteristics. To test this observation they were correlated between themselves. The results are presented in Table 11 for all data from propane through octanes and in Table 12 for octanes. Table 11. Correlation coefficients between the values of the largest eigenvalues of tested "mean degree of vertices" matrices for propane through octanes. Ax 1 Gx 0.99994 1 Lnx 0.99997 0.999993 1 Zgx 0.96 0.95 0.95 1 RAx 0.993 0.993 0.993 0.94 1 RGx* 0.994 0.994 0.994 0.94 0.999987 1 RLnx 0.994 0.993 0.993 0.94 0.999994 0.999998 1 RZgx 0.98 0.98 0.98 0.93 0.998 0.998 0.998 1 * In correlations with RG? the values obtained with the exponent c = -? were not included since RG?(-?) = 1. Table 12. Correlation coefficients between the values of the largest eigenvalues of tested "mean degree of vertices" matrices for octanes. Ax 1 Gx 0.99994 1 Lnx 0.99997 0.999993 1 Zgx 0.96 0.96 0.96 1 RAx 0.994 0.993 0.994 0.95 1 RGx* 0.994 0.9945 0.994 0.95 0.999987 1 RLnx 0.994 0.994 0.994 0.95 0.999994 0.999999 1 RZg. 0.985 0.985 0.98 0.948 0.998 0.998 0.998 1 * In correlations with RG? the values obtained with the exponent c = -? were not included since RG?(-?) = 1. The values for ethane were excluded since they are all equal to 1. For the same reason the data of RG?(-?) was excluded. In both tables are evident high correlations between the true "mean degree of vertices" indices, on the one hand between A?, G?, and Ln?, and on the other hand between RA?, RG?, and RLn?. Therefore, when testing the usefulness of the "mean degree of vertices" indices, only one of the three can be taken since they are mutually related. For example, according to the contents of Table 10, the indices of the group G? should be taken when the size of molecules is important, A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … Acta Chim. Slov. 2002, 49, 467-482. 481 whereas the indices of the group RLnx should be taken when only the contribution of branching is important. Conclusions The largest eigenvalues of the "mean degree of vertices" matrices tested here form a group of BIM type indices. They are derived from the matrix having the non-diagonal elements g$ (irj) = ("mean(v i ,vj )")xdij c and presented here as XYx(c) indices. The largest eigenvalues of the "mean degree of vertices" matrices, derived from the matrices having in their non-diagonal elements the exponent c = 0, are degenerated. The index RGx(-oo) is completely degenerated. When c < 0, the XYx(c) indices of n-alkanes have a declining increase of values with carbon number, with RZgx(c), which decreases with carbon number, as an exception. At c = 0, most tested XYx(c) indices are increasing either slightly more or slightly less than linearly. At c > 0, the increase can be described by higher polynomials. Among other isomers, on increase of the size of molecule also increase the values of most XYx(c) indices tested here. The exceptions are the indices RG^(c) when -oo < c < -2 and RZgx(c) when c < -2, where the values of indices for some isomers increase and for the others decrease. At RZgx(-2 and -1) all values decrease on increasing the size of the molecule. The sequence of isomers on increasing the size of molecule is regular at the index RGx(c) when -16 < c < -8. A regular sequence of isomers presenting only the information about the contribution of the number of branches and of the type of the branched structure have the indices Ax(0), Gx(0), Lnx(0), RAx(0), RG^(0), RLnx(c), and RZgx(0). Among most of tested XYx(c) indices the highest influence on the values of indices when the size of molecule increases, has the separation between branches. Lower than the influence of the separation between branches is in most cases the influence of the type of branches or of their number. Regular sequences of isomers of increasing branching have only the indices RAx(-1), RGx(-1 or -2), and RLnx(-1). The XYx(0) indices present only the information about the contribution of the number of branches and of the type of the branched structure. These indices are degenerated but true branching indices. A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of … 482 Acta Chim. Slov. 2002, 49, 467-482. The largest eigenvalues of the "mean degree of vertices" matrices perform in general better as indices for indexing the influence of the size of molecule, and less well as branching indices. To index the influence of the size of molecule the use of the indices G?(c) is recommended, and to index the influence of branching the indices RLn?(c). References 1. Wiener, H. J. Am. Chem. Soc. 1947, 69, 17–20. 2. Randić, M. J. Am. Chem. Soc. 1975, 97, 6609–6615. 3. Randić, M.; Basak, C.; Pompe, M.; Novič, M. Acta Chim. Slov. 2001, 48, 169–180. 4. Perdih, A.; Perdih, B. Acta Chim. Slov. 2002, 49, 67–110. 5. Perdih, A.; Perdih, B. Acta Chim. Slov. 2002, 49, 291–308. 6. Perdih, A. Acta Chim. Slov. 2000, 47, 293–316. 7. Bonchev, D.; Trinajstić, N. J. Chem. Phys. 1977, 67, 4517–4533. 8. Perdih, A.; Perdih, M. Acta Chim. Slov. 2000, 47, 231–259. 9. Lide, D.R. CRC Handbook of Chemistry and Physics, 76th Ed., CRC Press, Boca Raton 1995–96. 10. Dean, J.A. Lange's Handbook of Chemistry. McGraw-Hill, New York 1985. 11. Pogliani, L. J. Phys. Chem. 1995, 99, 925–937. 12. Ren, B. J. Chem. Inf. Comput. Sci. 1999, 39, 139–143. 13. Goll, E.S.; Jurs, P.C. J. Chem. Inf. Comput. Sci. 1999, 39, 1081–1089. 14. Gutman, I.; Ruscić, B.; Trinajstić, N.; Wilcox, C.F. J. Chem. Phys. 1975, 62, 3399–3405. Povzetek Med indekse tipa BIM spadajo tudi lastne vrednosti matrik, katerih elementi vsebujejo povprečne valence točk in potencirane razdalje med njimi. Ti indeksi dobro kažejo doprinos povečanja molekul, slabše pa doprinos razvejanja. Pravi, vendar degenerirani indeksi razvejanja iz te skupine so indeksi A?(0), G?(0), Ln?(0), RA?(0), RG?(0), RLn?(0) in RZg?(0). Regularno zaporedje izomer pri razvejanju imajo indeksi RA?(-1), RG?(-1 in -2) in RLn?(-1). A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 4. "Mean degree of …