Acta Chim. Slov. 2002, 49, 291-308. 291 SOME TOPOLOGICAL INDICES DERIVED FROM THE vmdn MATRIX. PART 2. THE "MEAN DEGREE OF VERTICES" SUMMATION-DERIVED INDICES OF THE BIM-TYPE Anton Perdih,* Branislav Perdih Mala vas 12, SI-1000 Ljubljana, Slovenia Received 03-09-2001 Abstract The "mean degree of vertices" indices tested here form a group of summation-derived indices of the BIM-type. One of them is the famous Randić index %. They are good indices for indexing the influence of the increase of the size of molecule, but less good as branching indices. The Randić index % (= RGE(-oo)) is one of the best of them. Better than it are in many cases other indices RGE, although the difference is not great in all instances. True but degenerated branching indices are the indices AE(-oo), GE(0), LnE(0), ZgE(0), RAE(0), RGE(0), RLnE(0), and RZgE(0). 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 = (v^vj )-1/2xdij -00, where v is the degree of vertex and dij is the shortest distance from vertex i to vertex j. In other words, the Randić index x is 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. 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 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). In 23M6 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 292 Acta Chim. Slov. 2002, 49, 291-308. 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.4,5 As criteria regarding the degree of branching is used the intuition6 as well as the Methane based definition and the n-Alkane based definition7 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 Handbook8 or from Lange's Handbook9. The data for the liquid molar volume (Vm), the intrinsic molar volume (Vi), the ratios Tc2/Pc and Tc/Pc used instead of the van der Waals parameters a0 and b0, BP/Tc, and the molar refraction (MR) were calculated from data presented in the handbooks. Results and discussion The "mean degree of vertices" matrix and the indices derived from it by summation The vmdn matrix4 is a matrix having the main diagonal elements, gii = 0, and all the others, g ij (i*j) = vj mVdi 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 this group of indices is one of the A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 293 most useful indices, the Randić2 index %, having the exponents a = -1/2, b = -1/2, c = -oo, whereas its zero order variant, the index \, 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 (i*j) = (v i^vj)-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 ,v j)", where g$ (irj) = ("mean(vi,vj)")xdij c. On the one hand, the indices were derived using as the "mean(v i ,v j)": - The arithmetic mean (average) of the degrees of vertices, (vi+vj)/2, the indices being labelled as As or As(c), - The geometric mean, (v^vj)1/2, the indices being labelled as Gs or Gs(c), - The logarithmic mean (v-vj )/(lnv-lnvj), the indices being labelled as LnE or Lns(c). For comparison, the Zagreb10 product (vixvj)2 was included as a "mean(v i ,vj)" and the indices derived using it were labelled as Zgs or Zgs(c). The indices were derived also using the reciprocal values of "mean(v i ,v j)" and they were labelled as RAE, RGS, RLnE, RZgE or RAE(c), RGs(c), RLnE(c), RZgE(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, are used when speaking about that group of indices as a whole, whereas the forms as e.g. As(c), are used when speaking about that index at the value of exponent c specified in parentheses. The general label for all these indices is XYs(c). These indices are summation-derived as presented by Eq. 1: XYs(c) = 1 ?("mean(vi ,v j )")xdij c (1) They obey the Methane-based definition of branching,7 therefore they belong to the BIM-type of indices. A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 294 Acta Chim. Slov. 2002, 49, 291-308. Characteristics of the "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 are the A? and Zg? indices when c = -? or an integer. The degeneration of the "mean degree of vertices" indices The number of degenerated data, i.e. of data having at least one equal counterpart among the "mean degree of vertices" indices of alkanes from propane through octanes, is presented in Table 1. Table 1. The number of degenerated data of the "mean degree of vertices" indices of alkanes from propane through octanes. Maximum possible number is 38. e Index -oo -oo < e < -2 0 1 2 3 As(c) 31 Gs(c), LnE(c) 4 Zgs(c) 4 RAe(c) 14 RGs(c) 4 RLns(c) 4 RZgz(c) 16 38 24 26 24 22 24 24 Table 2. The number of degenerated data of the "mean degree of vertices" indices of octanes. Maximum possible number is 18. c Index -oo -oo < c < -2 0 A? (c) 16 G?(c), Ln?(c), Zg?(c) 4 RA?(c) 8 RG?(c), RLn?(c) 4 RZg?(c) 2 18 15 15 15 15 The degeneration causing exponents are c = -oo and especially c = 0, whereas some degeneration is observed also at c = -1, 1 or 2. The most degenerated among tested indices is the index As, followed by RAE, RZgE, and ZgE. The least degenerated are the RGS indices, which include the Randić index x. Among the data of the "mean degree of 8 2 4 2 2 2 4 2 2 2 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 295 vertices" indices of octanes, Table 2, the situation is similar. In this case, the index RZg? is the least degenerated. The increase of values of the "mean degree of vertices" indices of n-alkanes due to the increase of size of the molecule How the values of the "mean degree of vertices" indices of n-alkanes from propane through octanes increase with increasing carbon number is presented in Table 3 for all tested indices and in Fig. 1 for the index A? as an example illustrating the situation among all of them. Table 3. The schematic presentation of the shape of the increase of values of the "mean degree of vertices" indices of n-alkanes from C3-C8 and of the degree of the simplest polynomial having r2 > 0.9995. c Index -oo -oo< c<-8 -6 -4 -2 -1 0 1 2 3 A?(c), G?(c), Ln?(c) / Zg?(c) / RA?(c), RG?(c), RLn?(c) / RZg?(c) / 3 * 2 2 2 3 3 4 2 )3 )3 2 3 3 4 * 2 2 2 3 3 4 4 4 )3 2 2 3 )3 /: Linear increase J: Parabolic increase *: Apparently linear; r2 > 0.9995 The digits: The degree of the simplest polynomial having r2 > 0.9995 2500 2000 1500 1000 500 0 30 20 10 L T ^: Fig. 1. The dependence of values of the index A? on exponent c and on carbon number. Right: The enlarged view on the lower part of the left graph. At c = -? they have a linear increase. At -? < c < -6 the increase is apparently but not truly linear. At c = 0 the increase is quadratic. In most other cases, the increase can c c A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 296 Acta Chim. Slov. 2002, 49, 291-308. be described by a higher polynomial. The exponents -qo < c < -6 can be used to decrease or eliminate the degeneration of indices without affecting seriously their dependence on carbon number. The increase of values of the "mean degree of vertices" indices due to the increase of the size of molecules of other alkane isomers How the values of the "mean degree of vertices" indices of other isomers increase with the size of a molecule is the most easily presented by the susceptibility of typical structural types for the increase in carbon number by one, Si,i+1, cf. ref.5 We consider here S7,8: Hp -> Oct, 2M6 -> 2M7, 3M6 -> 3M7, 3Et5 -> 3Et6, 24M5 -> 24M6, 23M5 -> 23M6, 22M5 -> 22M6, 33M5 -> 33M6, and 223M4 -> 223M5. Table 4. The increase of values of the "mean degree of vertices" indices on increase of the size of molecule. Index -oo < c < -4 -2 -1 0 1 2 3 As(c) Gs(c), LnE(c) Zgs(c) i i i i i i i T i T T T i T T T T T T T RAe(c) RGy(c), RLny(c), RZg T ¦(c) T T i i i i i i i i i t T =: The values increase equally among all isomers I: The values increase more among isomers having a less branched structure t: The values increase more among isomers having a more branched structure 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 all "mean degree of vertices" indices tested here. There is only one case, where the increase is equal among all isomers, Table 4. In other tested cases there is observed a higher increase of values of the "mean degree of vertices" indices either among less branched isomers or among more branched ones. In Table 5 is presented, which structural feature has the highest influence on the increase of values of the "mean degree of vertices" indices on increase of the size of the c A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 297 molecule. In cases marked "a", "a1", or "BI" the sequence of isomers seems regular as it would be expected for a good branching index. The first situation is observed at the indices Gs(0) and Lns(0), the second at the indices RAs(0), RGs(0), RLns(0), and RZgs(0), as well as at As(-oo). The last situation is observed at the index As(c) when -oo < c < -6 as well as at the index RZgs(-2). Among all indices tested here, the distance between branches has the highest influence when c > 0. This is the case also at the indices Gs, Lns, and Zgs when c < 0, as well as at the indices RAS, RGS, RLnS, and RZgE when c < -4. Table 5. The influence of structural features on the increase of values of the "mean degree of vertices" indices due to the increase of the size of molecule. c Index -00 -oo < c < -6 -4 -2 -1 0 1 2 3 As(c) Gz(c), Lns(c) Zgs(c) RAs(c), RLns(c) RGs(c) RZgE(c) a1 Dt Dt Dt Dt Dt BI Dt Dt Dt Dt Dt bri Dt Dt Dt Dt Dt b4 Dt Dt Dt b4 BI Dt Dt Dt b4 bri b4 a a2 a1 a1 a1 Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl Dl = : all values are equal a: 0 < 1t < 2t < 2q < 2q1t a1: 0 > 1t > 2t > 2q > 2q1t a2: 2q > 0 > 1t > 2q1t > 2t BI: 0 > 2M > 3M > 3Et > 24M > 23M > 22M > 33M > 223M br: The number of branches has the highest influence I The higher the number of branches, the lower the value of the index f The higher the number of branches, the higher the value of the index Dl: 24M < 0 < ... < 223M < 33M, 22M, i.e the influence of dist > Nbr D f: 33M, 22M < 0 < ... < 223M < 24M, i.e the influence of dist > Nbr The number of branches has the highest influence on the indices when the size of molecules increases, in the case of indices AE(-2), AE(-4), RAE(-1), RGS(-1), RGs(-2), RLnE(-1), and RZgE(-1). Since the values of the "mean degree of vertices" indices increase with the size of molecule, they are the BIM-type indices, i.e. they obey the Methane-based definition of branching.7 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 298 Acta Chim. Slov. 2002, 49, 291-308. The dependence of the "mean degree of vertices" indices on branching The increase or decrease with branching Whether the "mean degree of vertices" indices increase or decrease with branching is presented in Table 6. Table 6. The values of the "mean degree of vertices" indices increase or decrease with branching Index -? ? c ? -4 -2 -1 A?(c) G?(c), Ln?(c) Zg?(c) RA?(c), RG?(c), RLn?(c) RZg?(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 Most of them decrease when c > 0. The indices A?, G?, Ln?, and Zg? increase when c < 0, whereas most "reciprocal mean degree of vertices" indices decrease with branching when c < -2. Among the indices RZg? the values for some isomers increase and the values for the others decrease with increasing branching. The influence of structural features Which structural feature influences most strongly the values of the tested "mean degree of vertices" indices is presented in Table 7. In majority of cases it is the number of branches (b). The index RA?(-1) would be a candidate for a good branching index if it would not have a pair of degenerated values. The indices G?(0), Ln?(0), Zg?(0), RA?(0), RG?(0), RLn?(0), and RZg?(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. The indices A?, G?, Ln?, and Zg? present also the information that when branches c 0 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 299 are placed at or near the centre of molecule, then the structure seems to be more branched than if they are placed at the periphery of molecule. Table 7. Which structural feature influences the values of the "mean degree of vertices" _________indices the most strongly.__________________________________________ ________________________________c______________________ Index______________-od -oo < c <-4 -2 -1 0 1 2 3 AE(c) a2 b>e>c>s b>e>c>s b>e>c>s 0 b>e>c>s b>e>c>s b>e>c>s Ge(c), Lns(c) b>e=c>s b>e>c>s b>e>c>s b>e>c>s a1 b>e>c>s b>e>c>s b>e>c>s ZgE(c) b>s>e=c b>s>e>c b>s>e>c b>s>e>c a b>e>c>s b>e>c>s b>e>c>s RAs(c), RGe(c), b>e=c>s b>e>c>s b>e>c>s b>e>c>s a b>e>c>s b>e>c>s b>e>c>s RLns(c) RZgE(c) e=c>b>s e>c>b>s b>e>s>c b>s>c>e a b>s>e>c e>c>s>b b>e>c>s a: 0 < 1t < 2t < 2q < 3t < 2q1t < 4q a1: 0 > 1t > 2t > 2q > 3t > 2q1t > 4q a2: 0 < 1t < 2t < 2q = 3t < 2q1t < 4q b>e>c>s: The contribution of the number of branches (b) is higher than the contribution of ethyl (e) vs. methyl groups, followed by the position of branches (c: central) and the separation between branches (s) The "mean degree of vertices" indices that might be good branching indices Taking into account the information presented in Table 5 and 7 we can conclude that no one of the tested "mean degree of vertices" indices can be considered as a candidate for a good branching index. Namely, none of them has an ideal sequence of isomers when the size of molecule as well as when the branching increases. Correlation of physicochemical properties with the "mean degree of vertices" indices The correlation of the "mean degree of vertices" indices with the tested physicochemical properties assuming the linear relationship was derived using data of all alkanes from propane through octanes, as well as only for octanes. The former case is illustrated in Fig. 2 with help of the As index as a general example. The pattern observed testing the indices Gs and LnE is quite similar but not equal. The pattern observed on testing the index ZgE is less similar, since at c < 0 the correlations are lower. The pattern when testing the indices derived from the reciprocal values of the "mean(vi ,vj)" is less similar; therefore it is illustrated in Fig. 3 with indices RGS, RLnE, and RZgE. The pattern observed testing the index RAE is similar to that of RGS. A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 300 Acta Chim. Slov. 2002, 49, 291-308. m D D D D D D^J 0.8 l Î 0.6 >^ 0.4 0.2 0 -0.2 -0.4 -oo -16 -12 -10 -8 -6-4-2-1 0 1 2 3 A—a—a—a—a—& -0.6 ?—D—D—D—D -0.8 - -1 BP d Tc Pc -*V" Fig. 2. Correlation coefficient of values of the index A?(c) with tested physicochemical properties for all alkanes from propane through octanes. A quite different pattern is observed when only data of octanes are correlated. Two examples are presented here. One of them is the index A? presented in Fig. 4, the other one is the index RG? presented in Fig. 5. In graphs of the indices A?, G?, Ln?, and Zg? the curves cross the abscissa only once; at c = 0, c ? -1/2, c ? -1/2, and 0 < c < 2, respectively. In graphs of the indices RA?, RG?, RLn?(c), and RZg?, on the other hand, the curves cross the abscissa twice, cf. Fig. 5; namely at c ? -3 and c ? 1, at c ? -3 and c ? 1, at -16 < c < -10 and -6 < c < -4, as well as at -6 < c < -2 and 1 < c < 3, respectively. 1 kdc •Vc Hv Hf A B C Zc ?^w ac Tc2/Pc Tc/Pc Vm MR nD BP/Tc c A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 301 ^ ^ 0.4 0.2 - 0 -O.2-90 -16 -12 -10 -8 -6 -0.4 -0.6 - -0.8 -1 -4 -2 -1 0 1 2 3 1 0.5 0 -0.5 -1 J É=sgl=gN=B—B—Bh-=g -0o -16 -12 -10 -8 A A -6 A -4-2-10123 1 0-6 0-4 0.2 0 -0.2-co -0.4 -0.6 -0.8 -1 Fig. 3. Correlation coefficient of the values of the index RG?(c) and RZg?(c) with the tested physicochemical properties for all alkanes from propane through octanes. The symbols are the same as in Fig. 2. c c c A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 302 Acta Chim. Slov. 2002, 49, 291-308. D—D—D—D 0.8 0.6 4 * * 0.4 0.2 0 -oo -16 -12 -10 -8 -6 -4 -2 -1 -0.2 -0.4 -0.6 -0.8 -o-----o-----o-----o -1 A Xw BP d Tc Pc dc Vc Hv Hf A B C Zc ac Tc2/Pc Tc/Pc Vm MR nD BP/Tc V" Fig. 4. Correlation coefficient of values of index As(c) with tested physicochemical properties for octanes. In Fig. 5 we can see that the values of the correlation coefficients differ from one physicochemical property to another, as well as from one tested index to another 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 give rise to the best correlation of the particular index and physicochemical property, when the data of alkanes from propane through octanes are considered. The "best" exponents differ widely. It is interesting that the Randić index x (x = RGs(-oo)) correlates best in five of 21 cases (with oc, Zc, BP/Tc, co, and C). In the majority of other cases give rise to better correlations other negative exponents, e.g. c = -4. 1 c A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 303 1 0.8 0.6 0.4 0.2 0 - -0.4 -0.6 A------L------L------L------â------& -0.8 --1 Fig. 5. Correlation coefficient of values of the index RGs(c) with tested physicochemical properties for all octanes. As can be deduced from Fig. 2 and Table 8, these correlations are in many cases (BP, AHv, B, Tc2/Pc, Tc/Pc, and V"; Ar2 < 0.01) only marginally better than that of index x and do not contribute a substantial improvement. In other cases, the differences in correlation are higher: 0.01 0.1: AHf°g, dc, d, nD. 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. There is only one case in Table 9 where the Randić index x has the best correlation (AHv; r = 0.997), which is quite high. In spite of that, the situation is not unfavourable for the Randić A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 304 Acta Chim. Slov. 2002, 49, 291-308. index x, since Fig. 5 shows that in several cases, there is not a substantial improvement of correlation on replacing the exponent c = -oo, observed at the index x, for the "best" exponent: Ar2 < 0.01: Zc BP, AHf°g, A, and B 0.01 0.1: Vc, dc, Pc, oc, Tc, BP/Tc, Tc/Pc, Tc2/Pc, co, C, d, Vm, V", nD, and MR. In Table 10 are collected the cases of "best" correlations for data of all alkanes from propane 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.7 When all tested alkanes are considered, in 17 out of 21 cases | r | > 0.9 A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 305 and in five cases even | r | > 0.99. When only data of octanes is considered, then I r | > 0.9 is observed in 7 cases and | r | > 0.99 in 3 cases. Consequently, several of the tested "mean degree of vertices" indices are good indices of the influence of size of the molecules and only few of them indicate well also 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 co, Tc/Pc, and BP/Tc, followed by Tc2/Pc, C, Pc, and AHv. This is the case with the indices As(-1), Gs(-1), Gs(-2), followed by Gs(1), RAS(-1), RZgs(3), and RAs(-oo), 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 "mean degree of _________vertices" index, XYL(c), among octanes.______________________________ Physicochem. c property As(c) Ge(c) LnE(c) Zge(c) RAE(c) RGs(c) RLnE(c) RZge(c) BP -oo 0 0 0 -4 -4 -4 -oo AHv -oo 0 0 0 -oo -°°(x) -oo 1 AHf°g -oo 0 0 0 -oo -4 -4 -oo Tc 1 -1 -1 1 1 1 1 2 Pc 1 -1 -1 2 1 1 1 3 Vc 3 -1 3 2 1 1 1 3 dc 3 -1 3 2 1 1 1 3 oc 3 3 3 3 3 3 3 3 Zc -oo 0 0 0 -6 -6 -6 -1 BP/Tc -1 -2 -1 -2 -2 -2 -2 -2 Tc/Pc -1 -1 -1 -1 2 2 2 3 Tc2/Pc -1 1 1 0 0 0 -1 0 CO 1 -4 -2 -1 -1 -2 -2 -1 d -2 -4 -2 -oo -2 -2 -2 -2 Vm -1 -4 -2 -oo -2 -2 -2 -2 V" -1 -4 -2 -oo -2 -2 -2 -2 A 3 0 0 0 -4 -4 -4 1 B -oo 0 0 0 -4 -4 -4 2 C -4 -oo -oo 0 -1 -2 -2 -2 nD -2 -6 -4 -oo -2 -2 -2 -2 MR 1 -1 -1 1 1 1 1 3 None of the Lns indices belongs to the best indices of this group. Considering all 21 tested physicochemical properties, for indexing the influence of the size of molecules A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 306 Acta Chim. Slov. 2002, 49, 291-308. are the best indices RLns (12 cases), RGe (8 cases), followed by RAS (2 cases), Zgs (1 case). For indexing of branching are the best ones the indices from the group Zgs (6 cases), followed by RZgs (4 cases), RAS (4 cases), Gs (3 cases), RLns (2 cases), As and RGS (1 case each). Table 10. The cases of the best correlation coefficients (rbest) between tested physicochemical properties and the "mean degree of vertices" indices, XYs(c), among alkanes from propane through octanes as well as only among octanes. Physicochem. C3-C8 C8 property rbest index c rbest index c BP 0.994 RGS -4 0.862 RGS -4 AHv 0.997 RGe -8 0.939 RAS -oo AHf°g 0.990 RLnE -2 -0.873 RZgi -oo Tc 0.986 RGe -4 -0.794 RZgs 2 Pc -0.949 RGS, RLnE -4 -0.945 RZgi 3 Vc 0.986 RGe, RLns -4 0.845 RAS 1 dc 0.805 RLnE -1 -0.832 RAE 1 oc 0.951 RLnE -6 0.720 RLnE 3 Zc -0.765 RLnE -oo 0.658 Zgi 0 BP/Tc 0.966 RAS -6 -0.991 Ge -2 Tc/Pc 0.994 RLnE -4 -0.994 Ge -1 Tc2/Pc 0.997 RLnE -4 0.977 Ge 1 CO 0.970 RAE -oo -0.996 Ae -1 d 0.931 RLnE -1 0.836 ZgE -oo Vm 0.976 RGS -4 -0.846 ZgE -oo V" 0.846 RGe -6 -0.845 ZgE -oo A 0.560 RLnE 3 0.724 RLnE -4 B 0.984 RLnE -4 0.715 RZgs 2 C -0.969 RLnE -6 0.969 RAe -1 nD 0.909 ZgE -1 0.804 ZgE -oo MR 0.995 RGS -4 0.882 ZgE 1 Conclusions The "mean degree of vertices" indices tested here form a group of summation-derived indices of BIM-type. They are derived from the matrix having the non-diagonal elements g$ (irj) = ("mean(v i ,vj )")xdij c and presented here as XYs(c) indices. One of them is the famous Randić2 index x, which is identical with the index RGe(-oo). The "mean degree of vertices" summation-derived indices, derived from the matrices having in their non-diagonal elements the exponent c = -oo, and especially those A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" Acta Chim. Slov. 2002, 49, 291-308. 307 having c = 0, are degenerated. To avoid degeneration without affecting seriously the properties of an index having c = -qo, i.e. being derived from the adjacency matrix, the exponent -qo < c < -6 is to be used. To eliminate the degeneration in the case when c = 0. two measures are to be taken: using c slightly different from 0 and using the exponent on the degree of vertex slightly different from 1. When c = -qo, the tested indices exhibit a linear increase with the size of molecule. At -oo < c < -6 the increase is apparently linear, at c = 0 it is quadratic. In other cases it can be described by higher polynomials. The sequence of isomers on increasing the size of molecule is regular at the index As(c) when -oo < c < -6, as well as at the index RZgs(-2). 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 As(-qo), Gs(0), Lns(0), Zgs(0), RAs(0), andRGs(0). A regular sequence of isomers on increasing branching has none of tested indices. Only the information about the contribution of the number of branches and of the type of the branched structure to branching present the indices As(-qo), Gs(0), LnE(0), ZgE(0), RAs(0), RGz(0), RLns(0), and RZgs(0). These indices are degenerated but true branching indices. The "mean degree of vertices" indices are good indices for indexing the influence of the increase of the size of molecule, but less good as branching indices. The Randić index x is one of the best of them. In many cases other indices RGS are better than it, although the difference is not always great. If the best-known reference values5 are taken as a criterion, then branching is indexed quite well also by the indices AE(-1), Gs(-1), Gs(-2), followed by Gs(1), RAS(-1). 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. Acta Chim. Slov. 2000, 47, 293-316. 6. Bonchev, D.; Trinajstić, N. J. Chem. Phys. 1977, 67, 4517-4533. 7. Perdih, A.; Perdih, M. Acta Chim. Slov. 2000, 47, 231-259. 8. Lide, D.R. CRC Handbook of Chemistry and Physics, 76th Ed., CRC Press, Boca Raton 1995-96. 9. Dean, J.A. Lange's Handbook of Chemistry. McGraw-Hill, New York 1985. 10. Gutman, I.; Ruscić, B.; Trinajstić, N.; Wilcox, CF. J. Chem. Phys. 1975, 62, 3399-3405. A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices" 308 Acta Chim. Slov. 2002, 49, 291-308. Povzetek Med indekse tipa BIM spadajo tudi nekateri indeksi izvedeni z uporabo povprečnih valenc točk in potenciranih razdalj med njimi. Mednje spada tudi Randićev indeks %, ki je identičen indeksu RGE(-oo) in je eden najboljših v tej skupini. Boljši od njega so marsikdaj drugi indeks iz skupine RGE. Ti indeksi dobro kažejo doprinos povečanja molekul, slabše pa doprinos razvejanja. Pravi, vendar degenerirani indeksi razvejanja iz te skupine so indeksi AE(-oo), GE(0), LnE(0), ZgE(0), RAE(0), RGE(0), RLnE(0) in RZgE(0). A. Perdih, B. Perdih: Some Topological Indices from the vmd n Matrix. Part 2. "mean degree of vertices"