Acta Chim. Slov. 2001, 48, 241-246.
241
ON TOPOLOGICAL INDICES INDICATING BRANCHING
PART 4. THE SUSCEPTIBILITY FOR BRANCHING AS A SOURCE OF
RVa-TYPE REFERENCE VALUES OR OF BIa-TYPE INDICES
Anton Perdih
Mala vas 12, SI-1000 Ljubljana, Slovenia
Received 11-01-2001 Abstract
Branching indices (BIM) or reference values (RVM) obeying the Methane-based definition as well as those obeying the n-Alkane-based definition of branching (BIA or RVA) can be derived from the susceptibility for branching, Sy, of other topological indices or physicochemical properties.
Introduction
In a previous paper, the Methane-based définition of branching as well as the n-Alkane-based definition of branching was suggested. Later, the relation between the Methane-based definition of branching obeying branching indices (BIm) and the n-Alkane-based definition of branching obeying branching indices (BIa) was presented . On the other hand, the reference values (RVm) for evaluation of indices intended to be used as BIm indices, as well as the reference values (RVa) for evaluation of indices intended to be used as BIa indices were derived from data of physicochemical properties AHf°g, Tc/Pc, and BP/Tc. Subsequently, BIm and BIa indices were derived from 11 most popular topological indices and compared to the reference values mentioned above. Meanwhile, the susceptibility for branching, Sy, used initially to evaluate the suitability of physicochemical properties as references for branching indices as well as the suitability of topological indices as branching indices, has been found to be a possible source of RVa or BIa data and this possibility is presented here.
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...
242
Acta Chim. Slov. 2001, 48, 241-246.
Data
The structures of alkanes are presented in shorthand, e.g. «-Oct is n-octane, 223M5
is 2,2,3-trimethylpentane, 3Et2M5 is 3-ethyl-2-methylpentane, etc.
I The indices
We decided to take into account the group of the most frequently used indices and
some novel indices. Altogether 11 indices are used. The data for Wiener index (W), the
Hosoya3 index (Z), the Randić  index (%) were taken from Yang et al . The ID numbers
were taken from Randić  and the Schultz MTI index from Mihalić et al . The Xu index
was taken from Ren  , whereas the Schultz indices S (Sch-S) and TLFCIR(D) (Sch-TF)
were taken from Schultz and Schultz  . The indices IX\   , the largest eigenvalue of the
adjacency matrix    (X\), and the largest eigenvalue of the distance matrix (D) were
calculated from corresponding matrices.
The properties
The data for the boiling point (BP), the critical temperature (Tc), the critical pressure (Pc), as well as the standard enthalpy of formation for the ideal gas (AHf°g) were taken from CRC Handbook.14 BP/Tc and Tc/Pc were calculated.
The susceptibility for branching
The susceptibility for branching, S/y, is defined as the normalised difference of the data of properties or indices of alkanes of different degree of branching, Eq. 1.
Sy = Xy/X,--l                                                                                                (1)
where S y is the susceptibility for branching and X is a physicochemical property or a topological index. In present paper, i refer to data for «-octane and y refer to data of any octane, i.e. the S„j7- data are used.
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...
Acta Chim. Slov. 2001, 48, 241-246.
243
Results and discussion
The reference values RVm and RVa as well as the indices BIm and BIa can be derived in different ways. ' It has been dissuaded from the use of the RVa values and BIa indices derived by Eq. 2 because their values are too dependent on the magnitude of the source data:
Y*a = abs(X, - X„)                                                                                            (2)
where Y*a stands for any potential RVa or BIa and n stands for the n-alkane of the same carbon number, Nc.
There exists a possibility to eliminate the dependency on the magnitude of source data by introducing into Eq. 2 the normalization of data, cf. Eq. 3:
Y'a = abs(X,- - X„)/X„ = abs(X//X„ - 1) = abs(S„,y)                                                (3)
According to Eq. 3 a RV'a value or a BI'a index is nothing but the absolute value of the susceptibility for branching (in alkanes at the same carbon number), S„j, that has been used previously ' as an indicator of the suitability of a properly as a reference as well as of an index as a branching index. In order to ensure that Y'a derived in Eq. 3 becomes compatible with the relation Ym = (Nc - 1) + Ya, it is to be modified as follows, Eq. 4 and 5:
YA = (Nc - l)*abs(S„,y)                                                                                      (4)
YM = (7Vc-l)*(l + abs(S,v))                                                                              (5)
The correlation of RVa and BIa data for octanes derived with help of Eq. 4, where the reference values RVa are derived from data of AHf°g, Tc/Pc, and BP/Tc, whereas the BIa indices are derived from a set of 11 topological indices tested earlier, is presented in Fig.l for the topological index Sch-TF and the physicochemical property Tc/Pc, as well as in Table 1 for all the tested topological indices and physicochemical properties. According to Table 1, the best BIa index compared to the RVa value derived from AHf°g is that derived from the topological index Z. In the case of Tc/Pc as the reference properly it is that derived from the topological index MTI and in the case of BP/Tc it is that derived from XX\.
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...
244
Acta Chim. Slov. 2001, 48, 241-246.
1.2
0.8
0.6
0.4
0.2
00
BlA(Sch-TF)
O
2233M4
223M5 Q
24M6 22M600/   23M6
224M5 O
33M6 0/03Et2M5 O
34M6
233M5
3 3Et3M5 234M5
25M6 0
3Et6
MM7
3M7
2M7
n-Oct
0.2
0.4
0.6
RVa(Tc/Pc)
I                                                                                                                                        I
0.8                      1
Fig. 1. Correlation between the BIa index derived from the topological index Sch-TF and the RVa reference value derived from the physicochemical property Tc/Pc.
Table 1. Correlation coefficient (r) of the linear regression of BIa and RVa data of octanes.
RVA from AHf°g		RVa from Tc/Pc		RVa from BP/Tc	
BIa from	r	BIa from	r	BIa from	r
Z	0.8958	MTI	0.9841	AA]	0.9851
X	0.8486	Sch-TF	0.9832	h	0.9744
ID	0.7425	W	0.9829	Xu	0.9722
Sch-S	0.6654	D	0.9809	D	0.9692
kk\	0.5773	Xu	0.9804	Sch-TF	0.9689
h	0.5708	AAi	0.9738	W	0.9685
D	0.5267	h	0.9586	MTI	0.9612
Xu	0.5193	Sch-S	0.9370	Sch-S	0.9582
Sch-TF	0.5061	ID	0.8886	ID	0.9200
W	0.5059	X	0.7572	X	0.8187
MTI	0.4666	Z	0.6832	Z	0.7435
					
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...
Acta Chim. Slov. 2001, 48, 241-246.
245
Table 2. The n-Alkane based definition obeying reference values for branching indices, RVa, as well as branching indices, BIa, derived by Eq. 4 from data of physicochemical properties AHf°g, Tc/Pc, and BP/Tc, as well as from data of topological indices %, Sch-TF, and IX\, respectively.
Octane	RVa(ahps)	RVa(Tc/Pc)	RVa(BP/Tc)	BIa(7)	BIa(ScIi-TF)	BIa(xa.i)
«-Oct	0	0	0	0	0	0
2M7	0.2361	0.0882	0.0279	0.2577	0.2193	0.0751
3M7	0.1405	0.2084	0.0551	0.1897	0.3532	0.1092
4M7	0.1222	0.2206	0.0529	0.0610	0.4005	0.1192
3Et6	0.0787	0.3487	0.0844	0.1218	0.5370	0.1538
22M6	0.5466	0.3301	0.1002	0.6322	0.5791	0.2131
23M6	0.1841	0.4226	0.1117	0.4176	0.6224	0.2074
24M6	0.3681	0.3562	0.0990	0.4476	0.5817	0.1905
25M6	0.4763	0.2150	0.0615	0.5156	0.4491	0.1527
33M6	0.3920	0.5032	0.1583	0.5238	0.7563	0.2684
34M6	0.1532	0.5343	0.1394	0.3496	0.7117	0.2354
3Et2M5	0.0927	0.5571	0.1547	0.3496	0.7590	0.2459
3Et3M5	0.2192	0.7011	0.2222	0.4153	0.8942	0.3142
223M5	0.3920	0.6684	0.2140	0.7740	0.9362	0.3507
224M5	0.5269	0.5019	0.1648	0.8901	0.8036	0.2988
233M5	0.2684	0.7606	0.2474	0.7336	0.9861	0.3682
234M5	0.3021	0.6347	0.1849	0.6452	0.8509	0.2999
2233M4	0.5859	0.9313	0.3261	1.1878	1.1634	0.4741
Table 2 presents for octanes the RVa values derived from data of AHf°g, Tc/Pc, and BP/Tc, as well as the BIa indices derived from data of topological indices %, Sch-TF, and Xk\ as examples. The indices % and Sch-TF were the second best ones in Table 1. They are used in Table 2 instead of indices Z and MTI because the latter ones cannot be considered good indices for other reasonsJ.
Comparing in Table 2 the BIa to RVa we can see that the presented "best" BIa indices are more susceptible to branching than the reference values to them they correlate the best. An exception is BIa(x) compared to RVA(AHfg) at 4-methyl heptane.
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...
246
Acta Chim. Slov. 2001, 48, 241-246.
References
1.    A. Perdih, M. Perdih, Acta Chim. Slov. 2000, 47, 231 -259.
2.    A. Perdih, Acta Chim. Slov. 2000, 47, 293-316.
3.    A. Perdih, Acta Chim. Slov. 2000, 47, 435-452.
4.    H. Wiener,./. Am. Chem. Soc. 1947, 69, 17-20.
5.    H. Hosoya, Bull. Chem. Soc. Japan 1971, 44, 2332-2339.
6.    M. Randić,./. Am. Chem. Soc. 1975, 97, 6609-6615.
7.    L. Lovasz, J. Pelikan, Period. Math. Hung. 1973, 3, 175-182.
8.    Y.-Q. Yang, L. Xu, C.-Y. Hu,./. Chem. Inf. Comput. Sci. 1994, 34, 1140-1145.
9.    M. Randić,./. Chem. Inf. Comput. Sci. 1984, 24, 164-175.
10.  Z. Mihalić, S. Nikolić, N. Trinajstić,./. Chem. Inf. Comput. Sci. 1992, 32, 28-37.
11.  B. Ren,./. Chem. Inf. Comput. Sci. 1999, 39, 139-143.
12.  H.P. Schultz, T.P. Schultz, J. Chem. Inf. Comput. Sci. 2000, 40, 107-112.
13.  M. Randić, Acta Chim. Slov. 1997, 44, 57-77.
14.  D.R. Lide, CRC Handbook of Chemistry and Physics, 76th Ed., CRC Press, Boca Raton 1995-96.
Povzetek
Indekse razvejanosti (BIM) oziroma referenčne vrednosti zanje (RVM), ki sledijo metanski definiciji razvejanosti, kakor tudi indekse razvejanosti (BIA) oziroma referenčne vrednosti zanje (RVA) po «-alkanski definiciji razvejanosti lahko izvedemo iz topoloških indeksov oz. fizikokemijskih lastnosti s pomočjo občutljivosti za razvejanje, Sv.
A. Perdih: On topological indices indicating branching part 4. The susceptibility for branching as...