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ON TOPOLOGICAL INDICES INDICATING BRANCHING PART 3.
ASSESSMENT OF SOME INDICES FOR THEIR SUITABILITY TO
REPRESENT BRANCHING†
A. Perdih
Mala vas 12, SI-1000 Ljubljana, Slovenia
This paper is dedicated to the late Professor Drago Kolar Received 22-08-2000
Abstract
The susceptibility for branching, Si,j, the difference-normalised difference of data for octanes, Anori, as well as a number of additional criteria are used to judge the suitability of 13 topological indices as branching indices. The indices EAmax and J are not conform with the criterion Si,j. The other indices comply also to the criterion Anormi. Due to the non-linear increase of data for n-alkanes are the most inappropriate the indices Z, W, MTI in l 1. The most susceptible for branching are Z, D, W, MTI, Sch-S, and l 1, whereas the least susceptible are ll 1 and ID. The same sequences of increasing branching as Tc/Pc have the indices MTI, Sch-TF, W, Xu, D, and ll 1, with rTc/Pc around 0.98. No one tested index has the same sequences of increasing branching as DHf°g and BP/Tc. The most evenly distributed Anori data have the indices Sch-TF and D. Index D behaves similarly to W, but is not degenerated. Of BIM and BIA types of indices seem the most appropriate BIA(c) regarding RVDHfg, BIA(Xu) regarding RVTc/Pc and BIA(ll 1) regarding RVBP/tc. BIa indices should be included into equations of the Kamlet-Taft type.
Introduction
Several hundred topological indices have been developed and tested for their performance as branching indices or indices of substances' properties. They have been correlated with several physical, chemical, and biological properties of molecules and the interest in this has grown remarkably during the past years. Therefore, the study of branching indices remains important. Ideally, one should test all indices and correlate them with all properties. Recent studies, however, indicate that a limited number of indices may suffice for this purpose and that only few properties are suitable as references to assess the indices.
Mendiratta and Madan1 reported that besides the Wiener2 index (W) the most useful indices are the Hosoya3 index (Z), the Ran die4 index (c), and the Balaban5 index (J). The most popular branching index, the Wiener index, W, was used even to define
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molecular branching , although it was developed to determine the paraffin boiling points . Another important index is l 1, the largest eigenvalue of the adjacency matrix7. At most a dozen indices emerge as the best single characterisation of diverse physicochemical properties of octanes8.
On the basis of these findings was studied in a previous paper9 the suitability of 11 topological indices (further on: indices) (cf. also below the chapter Data) J, W, Z, D, MTI, Xu, ID, c, ll 1, EAmax, and l 1 as branching indices, as well as of 24 physicochemical properties (further on: properties) MON, BP, d, Vi, Vm, Vc, Tc, Pc, dc, Zc, ac, DHv, A, B, C, nD, MR, a0, b0, DHf°g, DGf°g, S, R2, and w, as reference properties for branching of alkanes was tested first by means of the Principal Component Analysis. It has been observed that most of properties and indices correlate highly with carbon number of alkanes although they are influenced also by branching. Of the influences of branching, assessed separately by the properties and by the indices, the most important is the number of branches, followed by the type of branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary, by the position of branches, i.e. whether they are central or peripheral, distant or adjacent, and the least influence has the shape of branched molecules, i.e. whether they are spherical, flat or elongated, as well as symmetric or asymmetric. The properties have been divided into intrinsic and interaction-dependent ones and it has been explained why the latter ones are not suitable as primary references for branching. Two definitions of branching have been presented, the Methane-based definition as a general definition and the n-Alkane-based definition as a special definition more familiar to chemists9.
Afterwards10, additional criteria that might enable a better assessment of branching were discussed, first of all the criteria to assess the suitability of physicochemical properties as references for branching indices. The susceptibility for branching, Si,j, the difference-normalised difference of data for octan es, An orm i, as well as a number of additional criteria were introduced to judge the suitability of 24 physicochemical properties of alkanes as references for branching. The most appropriate as primary references for branching regarding these criteria seemed to be DHf°g, Tc/Pc, and BP/Tc. They should be used simultaneously. The following properties, Tc2/Pc, w, and especially
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C were considered to be less appropriate. Other tested physicochemical properties were found as inappropriate measures of branching. From the data of DHf°g, Tc/Pc, and BP/Tc the reference values for branching indices obeying the Methane-based definition as well as for those obeying the n-Alkane-based definition of molecular branching were derived10.
In present paper we decided to study only the most frequently used indices and some recently presented topological indices (later on: indices), whereas the tests presented here can be applied also to other indices if needed. Our decision is based on the assumption that with this selection of indices no relevant information about the molecular structure contained in the information space of all indices is lost. The indices are tested in the same way as previously10 the physicochemical properties, as well as by means of the reference values derived from DHf°g, Tc/Pc, and BP/Tc.
Data
The indices
We decided to take into account the group of the most frequently used indices and some novel indices. Altogether thirteen indices are used. The data for Wiener2 index (W), the Hosoya3 index (Z), the Ran die4 index (c), the Balaban5 index (J), the Yang-Xu-Hu11 index (EAmax) (denoted in present paper as EA) were taken from Yang et al11. The ID numbers were taken from Ran die12 and the Schultz MTI index from Mihaliæ et al 13. The Xu index was taken from Ren14, whereas the Schultz indices S (Sch-S) and TLFCIR(D) (Sch-TF) were taken from Schultz and Schultz15. The indices ll 16, the largest eigenvalue of the adjacency matrix7 (l 1), and the largest eigenvalue of the distance matrix (D) were calculated from corresponding matrices.
The properties and the reference values for indices They were taken from a previous paper10.
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Methods
The structures are presented in shorthand, e.g. 223M5 is 2,2,3-trimethylpentane or 3Et2M5 is 3-ethyl-2-methylpentane.
The susceptibility of indices for branching is used as the Method No. 1 to assess their suitability as branching indices. It is defined as the normalised difference of the indices' values, Eq. 1,
Si,j = BI i /BIj – 1                                                                                                 (1)
where Si,j is the susceptibility for branching, BI is a (potential) branching index, i refers to the more branched structure and j refers to the less branched structure. Which one of the structures is more branched is concluded by intuition as presented in ref6 as well as by the Methane-based definition and n-Alkane-based definition9.
Two groups of Si,j data are used. In the first group, i refers to the structures having the same number and type of branches as j, but one carbon more. For example, in Soctane,heptane octane means any octane having the same number and type of branches as a heptane. In SOct,Hp i refers to n-octane (Oct) and j to n-heptane (Hp), or in S2m7,2M6 to 2-methyl heptane (2M7) and 2-methyl hexane (2M6), etc. In the second group of S i,j data, i and j refer to alkanes of the same carbon number; for example, Si ,Oct refers to any octane and n-octane, S2M7,Oct refers to 2-methyl heptane and n-octane.
To assess the data, the following criterion is applied as the criterion No. 1: A topological index might be useful as branching index, if the sign of all Soctane,heptane values is the same and that of all Si ,Oct values is the same as well. It is not useful if the sign among members within each group differs. The two groups of sign may or may not be equal.
As the Method No. 2, the difference-normalised difference of the indices' values for octanes is used, Eq. 2,
Anori = (BIi - BIOct)/(BI2233M4 - BIOct)                                                               (2)
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Criterion No. 2: A topological index might be suitable as branching index if all 0 < Anori < 1, and it is not suitable if any Anori < 0 or Anori > 1.
The criteria No. 1 and 2 serve to eliminate the indices, which are the most inappropriate candidates for good branching indices. The other criteria presented below serve to rank the remaining indices.
As the Method No. 3 we use the dependence of indices on carbon number expressed in different ways:
The Soctane,heptane compared to the corresponding S-values derived from data for carbon number, NC,
The correlation coefficient with NC, the rNC. Criterion No. 3: No clear-cut criterion could be set in this case.
As the Method No. 4 we use the estimate of the linearity of the increase of the values for n-alkanes with carbon number.
Criterion No. 4: The closer to linearity is its increase in n-alkanes with carbon number the better is an index as a branching index.
As the Method No. 5 we use the dependence of indices on branching. As a general dependence it is expressed by the Si ,Oct data as well as by the ratio Si ,Oct/ Soctane,heptane. As special indicators we use the following ratios: S234M5,Oct/S2233M4,Oct, S25M6,Oct/S22M6,Oct, and S34M6,Oct/S33M6,Oct compare the susceptibility of an index for the presence of tertiary carbons vs. that of quaternary ones in octanes. The ratio S25M6,Oct/S34M6,Oct indicates whether an index is more susceptible to peripheral or central position of branches.
Criterion No. 5 could not be clearly defined yet, but some useful information is given by this method.
As the Method No. 6 we use the comparison of the sequences of octanes having the same number of branches and the extent of information that can be derived thereby.
Criterion No. 6: The index having the sequences closer to those expected by intuition as well as to those of reference properties is better as a branching index.
As the Method No. 7 we use the evenness of Anorm i data. It is assessed by the differen ces of An ori, i.e. A(Anormi) max, A( Anori) min, as well as by their relative standard deviation (RSD). Besides the evenness as such, from these data can be deduced
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also the influence of the number of branches, of the (mutual) position of branches, and of the symmetry of molecules among octanes on the indices considered potentially useful as branching indices.
Criterion No. 7: The index having more evenly distributed Anorm i data is better as a branching index.
Results and discussion
Selection of topological indices
Method No. 1: The sign of the susceptibility of indices for branching, S i,,j Method No. 2: The difference-normalised difference, Anornii
,10
Like the physicochemical properties of alkanes , the indices can be assessed for their susceptibility for branching, too. Table 1 presents the grouping of indices considered here regarding the sign of Si,j. The positive sign of Si,j is consistent with the fact that the indices X1, Sch-S, and Sch-TF increase with carbon number as well as with branching; therefore they are perspective candidates for good branching indices.
Table 1. Grouping of indices regarding the sign of their susceptibility for branching.
Soctane,heptane
>i,Oct
Indices
All +	All +
All +	All -
All -	All +
All -	All -
Some +, some -	All +
A1, Sch-S, Sch-TF
W, D, Z, x, ID, M1, MTI, Xu
EA, J
The indices W, D, Z, %, ID, Xk1, MTI, and Xu increase with carbon number and decrease with branching like several properties accounted for previously10, therefore it is not surprising that they are good indices of a single or for few those properties, but not good branching indices. On increasing carbon number, the index EA increases among n-alkanes, but it decreases for a series of other isomers, e.g. 2Mi, 3Mi, 3Eti, 22Mi, 33Mi,
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23Mi, 223Mi, etc. (i > 3). The index J decreases in the cases of peripheral position of branches. These indices violate the Methane-based definition of branching and for this reason the index J5 and especially the index EA11 are not satisfactory as branching indices and will not be considered further. All the remaining indices comply also with the criterion No. 2, i.e. 0 < Anormi < 1.
Ranking the 11 remaining indices
Method No. 3: The dependence on carbon number
To test the mean dependence of tested indices on carbon number in alkanes, the series of the mean susceptibility for the increase in carbon number from 7 to 8 at the same type of branching, S0Ctane,hePtane, of its relative standard deviation, as well as the correlation coefficient of the increase of their values with carbon number are presented in Table 2. For comparison are indicated in the series also the positions of reference properties AHf°g, Tc/Pc, and BP/Tc (in parentheses). According to Table 2, the indices ID, %, Xu, Xk\, Sch-S, D, MTI, W, and especially Z are in average more susceptible for the increase in carbon number than the reference properties and NC, whereas Xi is less susceptible. The relative standard deviation indicates that the spread of values of different isomers is the highest at Xi and the lowest at ID. The correlation with carbon number is quite high, especially at Xki and ID. The index Sch-TF follows the reference properties quite close.
Table 2. Susceptibility of tested indices to the increase in carbon number.
_________________________________________________^ octane,heptane________________________________________________
Mean   0.65 >Z>W~MTI>D>Sch-S>?iAi~Xu>%~ID> (Tc/Pc) >Sch-TF~ Nc = 0.14 >
(AHf°g)>(BP/Tc) > Xi > 0.01 RSD    0.6 > A,i> (AHf°g)>(BP/Tc) > 0.1 >Z>y> (Tc/Pc) >D~Xu~Sch-TF~MTI~W>
_______Sch-S>Mii > 0.01 > ID_____________________________________________
Tnc       0.998 >?iAi>ID > 0.99 > Xu ~ Sch-TF> (Tc/Pc)>(AHf°g) > 0.98 >Sch-S>%> ______D> MTI> 0.95 >W> (BP/Tc) > Z > A,i> 0.85____________________________
Method No. 4: The increase of the values for n-alkanes with carbon number
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How the indices increase with carbon number in n-alkanes is presented in Fig. 1: Z > W > MTI >>> D > Sch-S >> U1 >x>Arc> Sch-TF > Xu > ID >> X1. The indices Z, W, and MTI increase strongly with carbon number therefore they should be considered inferior as branching indices. Intermediate seem in this respect D and Sch-S, whereas TCk1 and % follow NC more closely. The indices Sch-TF and Xu increase slightly slower, whereas ID and especially X1 increase slower than NC. From this point of view, the indices TCk1, %, Sch-TF, and Xu seem to be the best of them. With exception of %, these indices were developed recently14-16. It seems as if the authors (intuitively or silently) tended to develop indices approaching the linear increase with carbon number for the data of n-alkanes. Such a linear increase was suggested recently10.
Fig. 1. The increase of indices for n-alkanes with carbon number (data normalised to n-butane). Between NC and Xu is placed Sch-TF, marked with empty triangle.
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Method No. 5: The dependence on branching
The series of the mean susceptibility for the increase in branching at the same carbon number (among octanes) Si ,Oct and its relative standard deviation as well as the ratio of the mean susceptibilities is presented in Table 3. The most susceptible for branching at the same carbon number are in decreasing order Z, Sch-S, D, MTI, W, and X1. At this carbon number is more susceptible for branching than for the increase in carbon number only X1. The other indices are less susceptible for branching than for the increase in carbon number.
Table 3. Mean susceptibilities of tested indices for branching.
__________________________________________Si ,Oct__________________________________________
Mean 0.2 > Sch-S>A1> 0.10 >Sch-TF> (AHf°g) > 0 > (BP/Tc) > -0.02~ID>U1>
(Tc/Pc) >X>Xu > -0.1 > W~MTI~D>Z> -0.23 RSD   0.65 ~ % > (BP/Tc)> (AHf°g) > Z > (Tc/Pc) > ID > U1 > l1 ~ Xu > Sch-S > Sch-
TF~D~W~MTI ~ 0.44________________________________________________
______________________________________________Si,Oct i  Soctane,heptane_____________________________________________
Mean  10 > X1 >> 0.7 >Sch-TF > (AHf°g) > Sch-S > 0 > ID ~ U1 > W~MTI > Z > Xu
> (Tc/Pc) > x > -0.4 > D > (BP/Tc) > -1__________________________________
RSD - relative standard deviation
Instead of mean Si,j, the Si,j of the most branched octane, 2233M4, as well as of other ones having a typical substitution pattern may be used for ranking the indices according to their susceptibility for branching. These rankings are presented in Table 4. The ranking according to S2233M4,Oct indicates again that the indices Sch-S, Sch-TF, and X1 give rise to the same sign of Si,j as AHf°g, whereas ID, Xk1, Xu, %, MTI, W, D, and Z give rise to the same sign of Si,j as Tc/Pc and BP/Tc. The ranking according to |S2233M4,Oct| indicates that the least susceptible for branching among the indices taken in consideration are the Randiæ indices ID and Xk1 and that the absolute values of their susceptibilities for branching are between those of AHf°g and BP/Tc.
The ratios S234M5,Oct/S2233M4,Oct, S25M6,8/S22M6,Oct, and S34M6,Oct/S33M6,Oct indicate that it depends on the position of branches, i.e. whether they are placed on the pepripheral or on the more centrally placed carbons, which of the indices gives more importance to tertiary carbons compared to quaternary carbons than the others. The ratio S25M6,Oct/S34M6,Oct
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indicates that the indices Z and c give like DHf°g more importance to peripheral substitution, whereas the other ones, especially l1, give more importance to central substitution.
Table 4. The ranking of indices regarding some susceptibilities for branching and their ratios.
Criterion______________________________________Ranking_________________________
S2233M4,oct                      0.24> Sch-S>Ai > Sch-TF > (AHf°g ) > 0 > (BP/Tc)>ID>?iAi >-
0.1> (Tc/Pc)>Xu>% >-0.3> MTI>W>D > Z = -0.5 |S2233M4,oct|                    0.5 > Z > D>W>MTI>0.3 > Sch-S>Ai > %>Sch-TF>Xu >(Tc/Pc) _____________________> 0.1 > (AHf°g) > Mii>ID>(BP/Tc)__________________________
S234M5,Oct/S2233M4,Oct          0.8 > MTI > W>D > (Tc/Pc)>Xu > ID>?lAl>Al>0.6> Z >
%>(BP/Tc)>(AHf°g) S25M6,oct/S22M6,oct           0.9 > (AHf°g ) > ID>Z>% > Xu>D>(Tc/Pc) > MTI>W > AAl >
0.7 >Sch-S>(BP/Tc)> Ai S34M6,oct/S33M6,oct           (Tc/Pc)>Xu > 1.0 > MTI>D>W >Sch-TF> (BP/Tc)>?iAi>ID
_____________________>Sch-S> Ai > 0.7 > % > Z > (AHf°g)_________________________
S25M6,oct/S34M6,oct           3.2 > (AHf°g) >Z>% > 1 > ID>Sch-S> 0.7 > D>?iAi> Xu > Sch-_____________________TF> W > MTI>Ai > (BP/Tc) > (Tc/Pc) > 0.4__________________
Method No.6: The influence of branching of octanes on indices
To evaluate the influence of branching of octanes on indices, there are presented in Table 5 the sequences of octanes having the same number and type of branches. Several indices, i.e. MTI, Sch-TF, W, Xu, D, and ll1 have the same sequence of octanes as Tc/Pc but the correlation of data is not what would be desired.
The correlation coefficients for data of octanes are as follows:
rTc/Pc,octanes: MTI (0.984) ~ Sch-TF (0.984) ~ W (0.983) ~ Xu (0.981) ~ D (0.981) >ll1 (0.976) > l1 (0.965) > Sch-S (0.941) > ID (0.891) > c (0.762) > Z (0.686)
rBP/Tc,octanes: ll1 (0.985) > Xu (0.972) > D (0.969) ~ W (0.968) > MTI (0.961) > ID (0.920) > c (0.819) > Z (0.744) > Sch-S (-0.958) > Sch-TF (-0.969) > l1 (-0.974)
rDHf°g, octanes: Sch-S (0.665) > l1 (0.571) > Sch-TF (0.506) > 0 > MTI (-0.467) > W (-0.506) > Xu (-0.519) > D (-0.527) > ll1 (-0.577) > ID (-0.743) > c (-0.849) > Z (-0.896)
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Table 5. The sequences of octanes having the same number and type of branches (t -tertiary, q - quaternary). < - less branched than the next one. If not marked E ________for ethyl, the branch is methyl.______________________________________
Number and type of branches
Index or	l(t)	2(t)	2(q)	3(t) or l(t)+2(q)
property				
Tc/Pc and *	2<3<4<3E	25<24<23<34<3E2	22<33<3E3	224<234<223<233
BP/Tc	2<4<3<3E	25<24<23<34<3E2	22<33<3E3	224<234<223<233
DHf°g	3E<4<3<2	3E2<34<23<24<25	3E3<33<22	233<234<223<224
Sch-S	2<4<3<3E	25<24<23<34<3E2	22<33<3E3	234<224<233<223
li	2<3<4<3E	25<24<23<34<3E2	22<33<3E3	234<224<223<233
ID	2<3E<3<4	3E2<34<25<24<23	22<33<3E3	234<224<223<233
c	4<3E<3<2	34<3E2<23<24<25	3E3<33<22	234<233<223<224
Z	3E<3<4<2	34<3E2<23<24<25	3E3<33<22	234<233<223<224
* indices: MTI, Sch-TF, W,		Xu, D, lli		
The correlation coefficients for data of all alkanes from ethane through octanes are
different, mostly appreciably higher due to the prevalent influence of carbon number:
rTc/Pc: Xu (0.997) > lli   (0.995) > ID (0.990)~ c (0.990) > D (0.987) > MTI (0.972) > W (0.971) > Sch-TF (0.956) > Sch-S (0.949) > Z (0.915) » li (0.777)
rBp/TC: Xu (0.976) > c (0.969) ~ D (0.968) > ID (0.956) > lli (0.950) > MTI (0.930) ~ W (0.929) > Z (0.886) > Sch-TF (0.881) > Sch-S (0.857) > li (0.705)
rDH°g: Sch-TF (0.990) > Sch-S (0.977) > ID (0.974) ~ lli (0.974) > Xu (0.960) > c (0.931) > D (0.914) > MTI (0.910) > W (0.905) > li (0.897) > Z (0.811)
The conclusion would be that the indices MTI, Sch-TF, W, Xu, D, and lli represent Tc/Pc very well qualitatively but not quantitatively. The other reference properties are not represented well by these indices.
Method No. 7: The evenness of the Anorm i data
The evenness of the Anormi data is presented here as A(Anorm i)max, A(Anori)m;n, and the relative standard deviation of the Anormi data, RSD, Table 6. The A(Anorm i)max an d RSD should be as low as possible, whereas A (Anori)m;n should be as high as possible. In this respect, the indices tend to distribute the Anorm i data more evenly than the best reference properties.
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In the A(An ori)min data one weakness of some indices is indicated, namely their degeneration. Regarding the index degeneration, the data up to octanes show the degree of degeneration l 1 > Z > W > MTI > c, Sch-TF, whereas among octanes it is Z > W, MTI > l 1, c. Due to their degeneration, these indices do not seem to be appropriate ones. Having also this fact in mind, and regarding the evenness of the Anormi data, then among tested indices the best ones are D and Sch-TF.
Table 6. The sequence of indices regarding the evenness of the Anori data.
Evenness                         Sequence
A(Anori)max                   ID > c > Sch-S > (BP/Tc) > ll 1 > Xu ~ D > MTI > W > l 1
> Sch-TF > Z > (Tc/Pc) > (DHf°g)
A(Anori)min                   Xu > (Tc/Pc) > (BP/Tc) > D > ll 1 ~ Sch-TF > Sch-S > ID
> > 0 = (DHf°g), c , l 1, MTI, W, Z
RSD                                 (BP/Tc) > (Tc/Pc) > (DHf°g) > c > Z > ID ~ ll 1 > l 1 ~ Xu ______________________Sch-S > Sch-TF ~ D > W > MTI______________________
Additional remarks
The indices Z, W, and MTI increase strongly with carbon number, whereas ID and especially l 1 increase slower. For this reason they should be considered inferior as branching indices. The indices ll 1, c, Sch-TF, and Xu seem to be the best of them in this respect, whereas the indices D and Sch-S are intermediate. The indices c, Sch-TF, MTI, W, and especially Z and l 1 are degenerated. Thus, especially the indices Z, W, MTI, and l 1 do not seem good candidates for branching indices. The index D, the largest eigenvalue of the distance matrix, behaves in many aspects similar to W but has not any degenerate value up to octanes. There is the question, why D is not used instead of W. Most indices derived from the distance matrix have a weak point, i.e. they decrease on branching. This makes them less suitable as branching indices and very suitable as interaction dependent property indices.
The index l 1 is derived from the adjacency matrix and behaves accordingly, i.e. it increases with increasing carbon number but with a lower correlation and increases with branching. It presents some mean of the number of C-C bonds per carbon atom in alkane. It has the value of 1 for the structure containing only primary carbons (i.e. ethane). On
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increasing the carbon number, it asymptotically approaches the limiting value of 2 for structures containing only two primary and a number of secondary carbons. It asymptotically approaches the limiting value of 3 for structures containing also tertiary carbons, as well as the limiting value of 4 for structures containing also quaternary carbons. The simplest alkanes containing besides primary carbons also only one secondary or only one tertiary or only one quaternary carbon, have the value of l1 equal to the square root of the number of C-C bonds of the central carbon. Only the difference between the square root of the number of C-C bonds of the central carbon in such an alkane and the number itself is available for all the other more complex structures. Though, l1 indicates some interesting rules. For example, the b,y-dimethyl alkanes have l1 equal 2 regardless the size of alkane. For this reason l1 should not be disregarded, although as a branching index l1 is not what would be desired and additional indices are to be looked for, using criteria presented above.
BIM and BIA indices derived from tested indices
In a previous paper10, the reference values RV to assess branching indices were derived from the properties DHf°g, Tc/Pc, and BP/Tc. In the same way can be derived from tested indices the candidates for branching indices that might be useful as branching indices obeying the Methane-based definition of branching (BIM) or as branching indices obeying the n-Alkane-based definition of branching (BIA), Eq. 3 -5.
BIM = (NC-1)* Xi /Xn                                                                                        (3)
if X is increasing with branching, and
BIM = (NC - 1)* Xn/Xi                                                                                       (4)
if X is decreasing with branching, where X is a topological index, n means the n-alkane of the same carbon number, NC, and i means any alkane of the same carbon number. The BIA indices can be derived from BIM indices simply10:
BIA = BIM - (NC - 1)                                                                                          (5)
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Another possibility would be to derive the RVi values, as well as the BIM and BIA indices in the following way, Eqs. 6 - 9:
RVM = (NC-1) + sign(Pi-Pn)*(Pi-Pn) RVA = sign(Pi-Pn)*(Pi-Pn)
(6)
(7)
BIM = (NC-1) + sign(Xi-Xn)*(Xi-Xn) BIA = sign(Xi-Xn)*(Xi-Xn)
(8)
(9)
The data derived by Eq. 6 - 9 give equal or similar correlation coefficients, as those derived by Eq. 3 - 5, but their values are too dependent on the magnitude of the source data; therefore it is dissuaded from their use.
The BIM and BIA indices derived by Eq. 3 - 5 from the remaining 11 indices tested here were correlated with reference values derived from DHf°g, Tc/Pc, and BP/Tc. The results are presented in Table 7 - 9.
Table 7. Correlation coefficients of BIM and RVM data. Data for all alkanes from ethane through octanes are included.
BIM	RVM from	BIM	RVM from	BIM	RVM from
from	DHf°g	from	Tc/Pc	from	BP/Tc
c	0.9969	Xu	0.9983	ID	0.9998
ID	0.9967	ll 1	0.9972	ll 1	0.9997
ll 1	0.9964	l 1	0.9963	Xu	0.9955
Xu	0.9948	ID	0.9959	c	0.9920
l 1	0.9903	MTI	0.9945	Sch-TF	0.9898
Sch-TF	0.9889	c	0.9937	l 1	0.9894
W	0.9828	Sch-TF	0.9937	MTI	0.9838
D	0.9828	W	0.9930	W	0.9821
MTI	0.9824	D	0.9927	D	0.9816
Sch-S	0.9672	Sch-S	0.9686	Sch-S	0.9614
Z	0.9643	Z	0.9583	Z	0.9461
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Table 8. Correlation coefficient of BIA and RVA data. Data for all alkanes from ethane through octanes are included.
BIa	RVA from	BIA	RVA from	BIa	RVA from
from	DHf°g	from	Tc/Pc	from	BP/Tc
c	0.8920	MTI	0.9726	ll 1	0.9635
Z	0.8666	W	0.9644	D	0.9630
ID	0.8034	D	0.9633	W	0.9627
ll 1	0.7164	ll 1	0.9460	MTI	0.9626
Xu	0.7072	l 1	0.9423	l 1	0.9586
l 1	0.6968	Xu	0.9278	Xu	0.9398
D	0.6537	ID	0.8945	ID	0.9216
W	0.6435	Sch-TF	0.8136	Sch-TF	0.8195
Sch-TF	0.5895	Z	0.7742	Z	0.8131
Sch-S	0.5851	c	0.7611	c	0.8121
MTI	0.5847	Sch-S	0.5879	Sch-S	0.6151
The correlation between BIMs and RVMs is high, up to over 0.999. This is not surprising since the influence of the factor having the most important contribution, NC, was set equal. More demanding is the correlation between BIAs and RVAs, especially if only octanes are considered. In the case of BIAs and RVAs for all tested alkanes, there is highly reduced the contribution of NC, whereas if only octanes are considered, it is entirely eliminated and the results reflect only the contribution of the influence of branching. Thus, this latter case is to be taken as the critical one in assessing the "goodness" of an index if in the former ones the index seems "good". In this respect is in the case of RVDHf°g the best BIA derived from c, in the case of RVTc/Pc that one derived from Xu, and in the case of RVBP/Tc that one derived from ll1.
Some BIX indices (X stands for M or A) show a typical behaviour relative to the others: The correlation coefficient of BIX derived from ID decreases in all cases as the contribution of NC decreases. The same holds true for those derived from ll1 and l1 in the case of RVs derived from DHf°g and Tc/Pc but not BP/Tc, as well as for those derived from Xu, Sch-TF, and MTI in the case of RVDHf°g.
The transformation of topological indices into BIX indices does not change the degeneration of those derived from c and Sch-TF, it improves the situation with those
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derived from l1 and Z, and makes worse the situation at those derived from W and MTI. The indices Z and MTI are for this reason as well as the reasons indicated above not to be considered good indices, whereas W should be replaced by D.
Table 9. Correlation of BI8 and RV8 data (only data for octanes are included). BIM and RVM as well as of BIA and RVA data for octanes give rise to equal correlation coefficients.
BI8	RV8 from	BI8	RV8 from	BI8	RV8 from
from	DHf°g	from	Tc/Pc	from	BP/Tc
Z	0.8958	Xu	0.9780	ll 1	0.9836
c	0.8486	Sch-TF	0.9778	l 1	0.9725
ID	0.7425	W	0.9775	Xu	0.9699
Sch-S	0.6654	MTI	0.9767	D	0.9661
ll 1	0.5773	ll 1	0.9759	Sch-TF	0.9657
l 1	0.5708	D	0.9758	W	0.9652
D	0.5267	l 1	0.9631	MTI	0.9576
Xu	0.5193	Sch-S	0.9393	Sch-S	0.9565
Sch-TF	0.5061	ID	0.8919	ID	0.9192
W	0.5059	c	0.7665	c	0.8194
MTI	0.4666	Z	0.6874	Z	0.7433
What would a good branching index be like?
Ideal index should either closely follow the increase in carbon number (in agreement with the Methane-based definition9) or should have zero value for nonbranched alkanes (n-alkanes; in agreement with the n-Alkane-based definition9), it should increase with branching, give for each structure a different value, etc. It should not necessarily give good single-parameter correlations with certain physicochemical properties of compounds but it should represent the branching contribution in methods like, e.g. the linear solvation energy relationship of Kamlet-Taft17, the QSAR method of Katritzky et al.18, and other ones. In the Kamlet-Taft correlation the size of molecules is contained in Vi whereas branching is not explicitly included. Its contribution may be hidden in different other parameters or in residual values and on inclusion of a separate parameter describing only the influence of branching the choice and the contribution of
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the other ones should be re-evaluated. For example, besides Vi, also BIA should be included.
On the other hand, on inclusion of a branching index based on saturated hydrocarbons, there may be no need to consider the influence of double or triple bonds on branching index, since the influence of double or triple bonds is already accounted for in other parameters of the Kamlet-Taft equation, e.g. in p*, a, b and analogous ones. The inclusion of an additional parameter presenting the influence of packing might be important in some cases.
References
1.     S. Mendiratta, A.K. Madan, J. Chem. Inf. Comput. Sci. 1994, 34, 867-871.
2.    H. Wiener, J. Am. Chem. Soc. 1947, 69, 17-20.
3.    H. Hosoya, Bull. Chem. Soc. Japan 1971, 44, 2332-2339.
4.    M. Randiæ, J. Am. Chem. Soc. 1975, 97, 6609-6615.
5.    A.T. Balaban, Chem. Phys. Lett. 1982, 89, 399-404.
6.    D. Bonchev, N. Trinajstiæ, J. Chem. Phys. 1977, 67, 4517-4533.
7.    L. Lovasz, J. Pelikan, Period. Math. Hung. 1973, 3, 175-182.
8.    M. Randiæ, S.C. Basak, J. Chem. Inf. Comput. Sci. 1999, 39, 261-266.
9.    A. Perdih, M. Perdih, Acta Chim. Slov. 2000, 47, 231-259.
10.  A. Perdih, Acta Chim. Slov. 2000, 47, 293-316.
11.  Y.-Q. Yang, L. Xu, C.-Y. Hu, J. Chem. Inf. Comput. Sci. 1994, 34, 1140-1145.
12.  M. Randiæ,J. Chem. Inf. Comput. Sci. 1984, 24, 164-175.
13.  Z. Mihaliæ, S. Nikoliæ, N. Trinajstiæ, J. Chem. Inf. Comput. Sci. 1992, 32, 28-37.
14.  B. Ren, J. Chem. Inf. Comput. Sci. 1999, 39, 139-143.
15.  H.P. Schultz, T.P. Schultz, J. Chem. Inf. Comput. Sci. 2000, 40, 107-112.
16.  M. Randiæ, Acta Chim. Slov. 1997, 44, 57-77.
17.  M.J. Kamlet, R.M. Doherty, M.H. Abraham, Y. Marcus, R.W. Taft, J. Phys. Chem. 1988, 92, 5244-5255.
18.  A.R. Katritzky, T. Tamm, Y. Wang, M. Karelson, J. Chem. Inf. Comput. Sci. 1999, 39, 692-698.
Povzetek
Obèutljivost za razvejanost, Si,j, normalizirane razlike podatkov za oktane, Anorm i, in še vrsta dodatnih kriterijev je bila uporabljena za vrednotenje 13 topoloških indeksov. Indeksa EAmax in J dajeta v isti skupini obèutljivosti za razvejanost tako pozitivne kot negativne vrednosti, zato nista primerni merili razvejanosti. Drugi preizkušeni indeksi ustrezajo tej in tudi zahtevi za normalizirane razlike podatkov za oktane, Anorm i. Zaradi nelinearnosti narašèanja vrednost i za n -alkane so najmanj primerni indeksi Z, W, MTI in l 1. Na razvejanje so najbolj obèutljivi indeksi Z, D, W, MTI, Sch -S in l 1, najmanj pa ll 1 in ID.
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Isto zaporedje vpliva lege vej na vrednosti kot pri Tc/Pc imajo indeksi MTI, Sch-TF, W, Xu, D in ll 1, pri èemer je rTc/Pc okoli 0.98. Noben preizkušen indeks nima enakega vpliva lege vej na zaporedje vrednosti kot DHf °g in BP/Tc. Glede na enakomernost porazdelitve vrednosti Anori sta najboljša indeksa Sch-TF in D. Indeks D naj bi uporabljali namesto indeksa W, ker nima degeneriranih vrednosti, druge lastnosti pa ima zelo podobne ali boljše kot W. Od BIM in BIA indeksov izvedenih iz preizkušenih indeksov so videti najprimernejši: BIA(c) glede na RVDHf °g, BIA(Xu) glede na RVTc/Pc in BIA(ll 1) glede na RVbp/tc. BIa indeksi naj bi bili vkljuèeni v enaèbe kot so npr. Kamlet-Taftove.
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