Acta Chim. Slov. 2003, 50, 83-94. 83 SOME TOPOLOGICAL INDICES DERIVED FROM THE vmdn MATRIX. PART 6. SUMMATION-DERIVED DIFFERENCE TYPE INDICES OF BIA CLASS Anton Perdih,* Branislav Perdih Mala vas 12, SI-1000 Ljubljana, Slovenia Received 07-12-2001 Abstract The AWA(m,n) indices derived from the vmdn matrix form a different group of topological indices of BIA class than the sWA(m,n) indices. Some of the AWA(m,n) indices correlate with the physicochemical properties co, Tc2/Pc, C, BP/Tc, and AHv having | r | > 0.9. Introduction Several hundred topological indices have been developed and tested for their performance as branching indices or indices of substances’ properties.1'2 Some succesful cases are for example the Wiener index,3 the Randič index,4 the largest eigenvalue of the adjaceny matrix,5'6 etc. Important steps towards having better indices are also the VTI indices,7 several new molecular matrices and other approaches to derive new indices,8"20 as well as the development of a matrix that enables derivation of an infmite number of indices.21 The majority of these indices belongs to the so-called BIM class indices. The BIM class indices obey the Methane based defmition of branching.22 There has been proposed also the n-Alkane based defmition of branching, which is a subdefinition of the Methane based defmition of branching, expected to be more familiar to chemists working in this field. The indices obeying the n-Alkane based defmition of branching were labelled BIA class indices. Two sets of BIA indices have been tested recently, those derived from a set of most popular topological indices,23'24 as well as the sWA(m,n) indices,25 which are the susceptibilities for branching of the W(m,n) indices.26 The W(m,n) indices were derived by summation of the elements of the so-called vmdn matrix. In the present paper are evaluated the BIA indices derived directly from the vmdn matrix. This generalized matrix allows the derivation of an infmite number of indices and we present here the properties of 225 indices derived from it. A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 84 Acta Chim. Slov. 2003, 50, 83-94. Data and methods Notations The structures of alkanes are presented in shorthand, e.g. Hp is «-heptane, Oct is «-octane, 223M5 is 2,2,3-trimethylpentane, 3E2M5 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 (i.e. the number of branches, Nbr = 2) in 22M6 are positioned on a quaternary carbon (i.e. the number of branches on quaternary carbons, Nq = 2) placed on the periphery (per) of the molecule. The two branches in 23M6 and 25 M6 are positioned on tertiary carbons (i.e. the number of branches on tertiary carbons, Nt = 2). 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 alkanes Data for alkanes from propane through octanes were tested. Presented are the results obtained with octanes. 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 refractivity index (nD) were taken from the CRC Handbook27 or from Lange's Handbook.28 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 (i.e. reduced BP), and the molar refraetion (MR) were calculated from data presented in the handbooks. The AWA(m,n) indices The AWA(m,n) indices are topological indices derived from the so-called vmdn matrix26 by summing its elements separately on the right side of the main diagonal and separately on the left side of it, Eq. 1 and 2: A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... Acta Chim. Slov. 2003, 50, 83-94. 85 WR = Z(v7mxd„n)rlght (1) WL = Z(vSdAeft (2) where vy is the degree of vertexy (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 / to vertex j (in alkanes it is the smallest number of bonds between the carbons in question). The difference of the calculated sums of matrix elements is then defined as the AWA(m,n) index, Eq. 3: AWA(m,n) = wL - wR (3) Results In a previous paper26 we studied the W(m,n) indices, which were derived from the vmdn matrix by summation of its elements. They are the BIM class of indices, i.e. they obey the Methane based definition of branching. To derive the BIA class of W(m,n) indices, i.e. the indices obeying the n-Alkane based definition of branching and labelled here as WA(m,n) indices, we have several possibilities. One of them is to use the susceptibility for branching29 of the W(m,n) indices26 as a group of WA(m,n) indices, the sWA(m,n) indices.25 Another possibility is to derive the BIA class indices directly from the vmdn matrix by summing the matrix elements separately on the right side of the main diagonal and separately on the left side of it, Eq. 1 and 2. Their difference is then defined as a new BIA class index, the AWA(m,n) index, Eq. 3. The choice of difference in Eq. 3 is arbitrary. There could also be wR - wL. In any čase are some indices negative. Later in the paper we shall show that the absolute value of the difference should be defined as theAWA(m,n) index, Eq. 4: AWA(m,n) = abs(Z(vymxd/)ieft - I(vymxd!yn)nght) (4) Characteristics o/AWA(m,n) indices The AWA(m,n) indices of «-alkanes are equal to zero since the matrices presenting their structure are symmetric. The AWA(m,n) indices of other alkanes are equal to zero when m = 0 for the same reason. The AWA(m,n) indices are integers only when n = -oo A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 86 Acta Chim. Slov. 2003, 50, 83-94. or 0 or an integer and m = -oo or an integer. As a consequence of their defmition, the sign of the AWA(m,n) indices is the same as the sign of the exponent m. Other properties of the AWA(m,n) indices are presented below taking octanes as examples. DegeneratedAWA(m,n) indices Ali AWA(0,n) indices are degenerated for reasons presented above. Highly degenerated are the AWA(-oo,n) and the AWA(m,-oo) indices. Few AWA(m,n) indices are degenerated when m = 2 or 3 and n = -2 or -1 or 1 or 2. How the values o/AWA(m,n) indices depend on exponents m and n 2500 1-2000 1500 -1000 I-500 0 -500 Fig. 1. The values of AWA(m,n) indices of 2,2,3,3-tetramethvl butane in the plain of exponents m and n. DWA = AWA(m,n). Fig. 1 serves as an illustration that the dependence of values of the AWA(m,n) indices on exponents m and n is train-like, whereas that of the sWA(m,n) indices25 has been shown to be saddle-like. The values of AWA(m,n) indices are high at high m and high n where they are for several orders of magnitude higher than the values of the sWA(m,n) indices. The comparison of values of the AWA(m,n) indices of some octanes, e.g. 2M7 > 234M5 > 223M4 > 4M7 > etc, suggests that some structural features greatly influence the values of these indices. For example, when m < 1 and n = 3, the most A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... Acta Chim. Slov. 2003, 50, 83-94. 87 different from the value of «-octane is the abs(AWA(m,n)) of 2-methyl heptane. When m = -6 and n = 2 this is true for 2,5-dimethyl hexane, when m > 2 and n = 3 as well as when m > -6 and n = 1 or 2 this is true for 2,2-dimethyl hexane, when m < 4 and 0 < n <1 this is true for 2,2,4-trimethyl pentane, when -oo < m < -4 and n ~ 0 this is true for 2,3,4-trimethyl pentane, when -2 < m < 1 and 0 < n < 1 this is true for 2,2,3-trimethyl pentane, when m < -2 and -K n < 0 this is true for 2,3,3-trimethyl pentane, and in other cases this is true for 2,2,3,3-tetramethyl butane. On the other hand, the value of abs(AWA(m,n)) the least different from the value for the structure of «-octane has 4-methyl heptane when m > 1. The same is true for 3-ethyl-2-methyl pentane when m < -2 and n > 1, whereas in other cases, when 0 * m * 1 as well as when n * -oo, this is true for 3-ethyl hexane. How the structural features influence the AWA(m,n) indices When we studied the physicochemical properties of alkanes we have shown that they are influenced by several structural features, e.g. by the size of molecule, by the number of branches, by the type of the branched structure, by the position of branches, by the separation between branches, etc.22 By definition (see Eq. 3 and 4) ali AWA(m,n) indices of «-alkanes are equal to zero. Thus the direct influence of the size of molecule is excluded. A qualitative presentation of the influence of particular structural features on the AWA(m,n) indices, based on sequences of isomers, is presented below. Number of branches Due to the definition in Eq. 3, the AWA(m,n) indices have without exception the same sign as the exponent m. If the position of the sum of the right part and of the left part of the vmdn matrix would be interchanged then also the dependence of the sign of the AWA(m,n) indices on the sign of the exponent m would reverse. The sign observed using Eq. 3 has thus no fundamental importance. So the absolute values of the indices, abs(AWA(m,n)), calculated by Eq. 4, can be used as indices. Therefore, from this point on we consider only the absolute values of the AWA(m,n) indices. A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 88 Acta Chim. Slov. 2003, 50, 83-94. Several octane isomers are presented by the AWA(m,n) indices as more branched than 2,2,3,3-tetramethyl butane, which is the most branched octane. The extreme situation among the tested indices is at the index AWA(-oo,3) where 15 out of 18 octanes are presented as being more branched than 2,2,3,3-tetramethyl butane. This fact cannot be explained by the contribution of the number of branches. It shows, on the other hand, that the AWA(m,n) indices and the sWA(m,n) indices25 are two different groups of the BIA class indices. Position of branches Regarding the influence of the position of branches we observe two cases. In a triangle at m < 1 and n < 0, there the octanes having the branches at or near the centre of molecule are presented by the AWA(m,n) indices as more branched than those having the branches at the periphery of the molecule, but the difference is small. The reverse is true at higher m and n, and the difference increases with m and especially with n being the highest at AWA(3,3). A transition region seems to exist between these regions. Some AWA(m,n) indices of octanes do not depend on the position of branches. These are the AWA(0,n) indices, the AWA(m,-oo) indices, and the AWA(-oo,0) index. Separation between branches Regarding the contribution of the separation between branches is the situation among the AWA(m,n) indices quite simple. The separation between branches does not influence the values of the AWA(m,-oo), AWA(-oo,n), and AWA(0,n) indices. The other AWA(m,n) indices indicate that an octane having a larger separation between branches is less branched. The highest influence in this direction is observed at high m and high n. Type of branches There are also some differences when the branches are of different type, i.e. whether a branch is methyl or ethyl. The indices AWA(m,-oo), AWA(0,n), as well as the index AWA(-oo,n) do not distinguish the influence of methyl from that of ethyl group. In majority of AWA(m,n) indices is presented the structure containing an ethyl group as less branched than a similar one containing a methyl group. Only at m > 1 and -oo < n < -1 A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... Acta Chim. Slov. 2003, 50, 83-94. 89 the contrary is the čase. The reason for this is not clear at the moment but the difference in the degree of symmetry in molecules as well as in matrices might be expressed during derivation of AWA(m,n) indices since the symmetric parts of the matrices are eliminated during subtraction. AWA(m,n) indices having special characteristics The values of indices AWA(0,n) are equal to zero. The index AWA(-oo,-oo) contains only the information about the number of branches. The indices AWA(m,-oo), when m * -oo or m * O, present the information that a structure containing a vertex of degree four is more branched than a structure containing two vertices of degree three (when -oo < m < 1), or equally branched as a structure containing three vertices of degree three (when m = 1) or even more branched than a structure containing three vertices of degree three (when m > 1). The indices AWA(-oo,n), when n > -oo, contain the information about the number of branches, the position of branches as well as the type of branches. Which sWA(m,n) indices could be good branching indices? There do not exist agreed criteria what characteristics should have a good branching index. Consequently, there exist several definitions of branching. One of the defmitions is, for example, based on the Wiener3 number,30 some other ones are based on the leading eigenvalue of adjacency matrix,5'6 or on the leading eigenvalue of path matrixu We presented the Methane based defmition of branching as well as the n-Alkane based one22 and concluded that branching should not be defined by topological indices but by structural features of molecules as well as that a good branching index should have a regular, possibly an "ideal" sequence of ali isomers.25 In a regular sequence of isomers the influence of the number of branches should be higher than that of the position of branches, followed by the separation between branches and the type of the branched structure. An “ideal” sequence of octanes of increasing branching would be, e.g., Oct < 2M7 < 3M7 < 4M7 < 3E6 < 25M6 < 24M6 < 23M6 < 34M6 < 3E2M5 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4, A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 90 Acta Chim. Slov. 2003, 50, 83-94. which indicates that a centrally substituted alkane is more branched than a peripherally substituted one, or, Oct < 3E6 < 4M7< 3M7 < 2M7 < 3E2M5 < 34M6 < 23M6 < 24M6 < 25M6 < 3E3M5 < 33M6 < 22M6 < 234M5 < 233M5 < 223M5 < 224M5 < 2233M4, which indicates that a peripherally substituted alkane is more branched than a centrally substituted one. The sign can be either < as above, or >. Among the AWA(m,n) indices, only few ones have a potentially regular sequence of isomers, i.e. those having -4 < m < V3 and n < -2. Their sequences are: n = -6, m = -2, -1; n = -4, m = -l: Oct < 3E6 < 2M7 < 3M7 < 4M7 < 25M6 < 3E2M5 < 24M6 < 23M6 < 34M6 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4 n = -6, m = -72, -V3, -V4, +V4; n = -4, m = -V2, -V3, -74: Oct < 3E6 < 2M7 < 3M7 < 4M7 < 25M6 < 24M6 < 3E2M5 < 23M6 < 34M6 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4 n = -4, m = -2: Oct < 3E6 < 2M7 < 3M7 < 4M7 < 25M6 < 3E2M5 < 24M6 < 23M6 < 34M6 < 22M6 < 3E3M5 < 33M6 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4 The AWA(m,n) indices have thus a lower potential to be good branching indices than the sWA(m,n) indices25 do. Correlation between the AWA(m,n) indices and physicochemical properties The intention of this paper is to indicate, which combinations of exponents' m and n values indicate the region of best correlation between the index values and the values of a number of physicochemical properties of octanes assuming a linear relationship. Fine-tuning of exponents to find the very best čase is beyond the scope of present work, therefore only one indicator, the correlation coefficient is used to indicate the region of best exponents' combinations. Only five of 20 tested physicochemical properties correlate with | r | > 0.9 with tested AWA(m,n) indices of octanes, Table 1. Ali of them except AHv belong to those physicochemical properties considered as the best available primary references for branching.29 In ali cases in Table 1 the exponent n is negative, whereas the exponent m may be either positive or negative, depending on the physicochemical property in question. If we compare in this respect the sWA(m,n) indices25 and the AWA(m,n) indices, then we can see that in most cases the sWA(m,n) indices correlate with tested physicochemical properties at least slightly better than the A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... Acta Chim. Slov. 2003, 50, 83-94. 91 AWA(m,n) indices. The exceptions are the physicochemical properties A (r = 0.80), B (r = -0.74), and Te (r = -0.72). Table 1. Combinations of tested exponents in the AWA(m,n) indices that give rise to the best correlations with values of selected physicochemical properties.___________________________ __________m,n r______m,n r______m,n r______m,n r______m,n r ra V2,-4 -0.975 V3,-4 -0.975 1,-4 -0.974 74,-4 -0.974 72,-6 -0.973 BP/Tc 1,-2 -0.930 V2,-4 -0.928 V3,-4 -0.928 V4,-4 -0.928 72,-2 -0.928 Tc2/Pc -1,-4 0.969 -1,-6 0.969 -2,-4 0.969 -l,-oo 0.969 -2,-6 0.968 C 1 -oo 0.965 1,-6 0.965 1,-4 0.964 V2 -oo 0.963 72,-6 0.962 AHv V^-oo -0.915 73,-00 -0.915 V2,-oo -0.914 -74,-oo 0.914 -73,-oo 0.913 The Information presented by the AWA(m,n) indices If we consider the information presented by the AWA(m,n) indices we should start with the indices having the most reduced information content, the AWA(0,n) indices. These indices present no information since AWA(0,n) = 0 for ali alkanes taken into consideration. Next to them is to be considered the index AWA(-oo,-oo). It bears the information about the number of branehes, Nbr, in the strueture of the alkane, which is the most fundamental information regarding branehing. This index can be considered as the most simple or primitive or degenerated, but a true branehing index presenting only the most important contribution to branehing of alkanes, cf. al so ref26 The simplest branehing index, AWA(-oo,-oo), is a member of two groups of indices, AWA(m,-oo) and AWA(-oo,n). For the indices of the group AWA(m,-oo) it is easy to show that they are, Eq. 5, AWA(m,-oo) =^x(4m-l) + Mx(3m-l) 5 showing clearly that one braneh on a quaternary carbon contributes more than one braneh on a tertiary carbon. Thus, the indices AWA(m,-oo) bear the information of the number of branehes (Nbr = Nq + Nt) as well as of the type of the branehed strueture, i.e. whether the braneh bearing carbons are tertiary (t) or quaternary (q), but nothing else. The information contained in indices AWA(-oo,n) is not as straightfonvard. For 3-ethyl-3-methyl pentane and 2,2,3,3-tetramethyl butane it can be shown that AWA(-oo,n) = -(Nbr +2n+2). For the other isomers only some qualitative conclusions can A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 92 Acta Chim. Slov. 2003, 50, 83-94. be drawn at the moment. The type of the branched structure, i.e. whether the branch bearing carbons are tertiary or quaternary, as well as the separation between branches do not influence the AWA(-oo,n) indices. On the other hand, isomers having equal number of branches as well as some similarities in the position of branches (equal distances of branches from the centre of the main chain regardless the side they are placed) have equal AWA(-oo,n): 22M6 and 25M6, 23M6 and 24M6, 33M6 and 34M6, 223M5 and 234M5. At the same number of branches, when n > 0, peripherally placed branches give rise to higher values of AWA(-oo,n) than the centrally placed ones, etc. The reverse is trne when n < 0. Ethyl substituted isomers have in any čase lower AWA(-oo,n) values than methyl substituted ones. This is most easily seen at the AWA(-oo,0) index, where it is also evident that an ethyl group placed on a tertiary carbon has a different contribution than on a quaternary carbon. The series of isomers of indices AWA(-oo,n>3): 0=Oct < 3Et2M5 < 3Et3M5 < 2233M4 < 3Et6 < 233M5 < 4M7 < 33M6 = 34M6 < 223M5 = 234M5 < 224M5 < 3M7 < 23M6 = 24M6 < 22M6 = 25M6 < 2M7 indicates that when n > 3, then the equivalent position of branches relative to the centre of the molecule gives rise to equal values of AWA(-oo,n). Discussion The AWA(m,n) indices considered here were found to be less suitable as branching indices as well as in most cases also less suitable as indices of physicochemical properties than the sWA(m,n) indices,25 which are the susceptibilities for branching24'29 of the W(m,n) indices.26 These groups of indices differ appreciably also in other characteristics. For this reason it can be reasonably expected that the AWA(m,n) indices may be useful in combination with the sWA(m,n) or other indices. There is, on the other hand, also the question, whether the present "one spot" test is representative for ali other possible indices of the AWA(m,n) type. The AWA(m,n) indices considered here were derived from the vmdn matrices, which have the main diagonal elements gu = 0 and the nondiagonal elements g$ (frj) = vymxd!yn In present čase, the vymxdn elements of the vmdn matrices are summed separately on the right side of the main diagonal and separately on the left side of it. The A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... Acta Chim. Slov. 2003, 50, 83-94. 93 absolute value of the sums' difference is then defined as a new BIA class index, the AWA(m,n) index. This way of index derivation is useful only when the matrices are not symmetric, which is the čase with the vmdn matrix mentioned above. If we consider a more general matrix,26 having the nondiagonal elements g$ (frj) = v,axvbxd/, then in the vmdn matrix a = 0, b = m, and c = n. The matrix having the nondiagonal elements g^ (&j) = V/axvybxd(,-c, will be nonsymmetric when a * b, which is the čase using the vmdn matrix presented above. When a * b, there is in fact given different weight to the contribution of v, than to vy. Another way to give different weight to these contributions is to consider the matrices having the nondiagonal elements g$ (frj) = (k,xvi)ax(kJxvJ)bxdIJc, where k, and ky are constants. Whenever k, * ky and/or a * b, the matrices are not symmetric and they are thus useful to derive the indices of the AWA(m,n) type. From this consideration follows that it is possible to derive a large number of sets of the AWA(m,n) indices and prior to concluding that they are ali inferior to sWA(m,n) or other indices, at least some additional sets of them are to be tested. A clue to properly choose the values of constants k, and ky, as well as of exponents a, b, and c, may be as follows: c should be negative and a should not be very different from b. By analogy, k, should not be very different from ky, but this assumption is stili to be tested. References 1. Randič, M.; Basak, C; Pompe, M.; Novič, M. Acta Chim. Slov. 2001, 48, 169-180. 2. Todeschini, R.; Consonni, V. Handbook of Molecular Descriptors, Wiley-CH, New York 2000. 3. Wiener, H. J. Am. Chem. Soc. 1947, 69, 17-20. 4. M. Randič, J. Am. Chem. Soc. 1975, 97, 6609-6615. 5. Lovasz, L.; Pelikan, J. Period. Math. 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Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6... 94 Acta Chim. Slov. 2003, 50, 83-94. 20. Randić, M.; Balaban, A. T.; Basak, S. C. J. Chem. Inf. Comput. Sci. 2001, 41, 593–601. 21. Estrada, E. Chem. Phys. Lett. 2001, 336, 248–252. 22. Perdih, A.; Perdih, M. Acta Chim. Slov. 2000, 47, 231–259. 23. Perdih, A. Acta Chim. Slov. 2000, 47, 435–452. 24. Perdih, A. Acta Chim. Slov. 2001, 48, 241–246. 25. Perdih, A.; Perdih, B. Acta Chim. Slov. 2002, 49, 497–514. 26. Perdih, A.; Perdih, B. Acta Chim. Slov. 2002, 49, 67–110. 27. Lide, D.R. CRC Handbook of Chemistry and Physics, 76th Ed., CRC Press, Boca Raton 1995–96. 28. Dean, J.A. Lange's Handbook of Chemistry, McGraw-Hill, New York 1985. 29. Perdih, A. Acta Chim. Slov. 2000, 47, 293–316. 30. Bonchev, D.; Trinajstić, N. J. Chem. Phys. 1977, 67, 4517–4533. Povzetek Indeksi vrste AWA(m,n), izvedeni iz matrik vmdn, tvorijo drugačno skupino topoloških indeksov tipa BIA kot indeksi sWA(m,n). Nekateri od indeksov AWA(m,n) korelirajo s fizikokemijskimi lastnostmi co, Tc2/Pc, C, BP/Tc inAHv z|r| > 0.9. A.Perdih, B. Perdih: Some Topological Indices Derived from the vmd n Matrix. Part 6...