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SOME TOPOLOGICAL INDICES DERIVED FROM THE vmd" MATRIX.
PART 8. THE Lij(m,n) INDICES
Anton Perdih,* Branislav Perdih
Mala vas 12, SI-1000 Ljubljana, Slovenia
Received 12-06-2002
Abstract
The best correlation between some Ljj(m,n) indices and physicochemical properties of alkanes, using data from propane to ali octanes inclusive is observed at the pairs Lij(0,0) and Mw (r = 1), LijC1^,1^) and Tc2/Pc (r = 0.9995), L1:(0,0) and MR (r = 0.998), Lij(0,72) and Tc/Pc (r = 0.998), L^A^O) and AHfg (r = 0.994), 1^(0,7:,) and Ve (r = 0.991), and Lij(V3,0) and BP (r = 0.991). The best correlation using only data of octanes is observed at the pairs Lij(72,-1) and Tc/Pc (r = -0.996), Lij(-74,-l) and <n (r = -0.994), 1^(1,-1) and BP/Tc (r = -0.991), Lij(74,3) and MON (r = -0.984), Lij(74,74) and Tc7Pc (r = 0.982), Li:(-72,-6) and C (r = 0.967), Li:(-l,-2) and S (r = -0.961), as well as at 1^(1,72) and Pc (r = -0.961). Thirteen Lij(m,n) indices having -1 < m < 0 and -2 < n < 0 have a regular sequence of isomers. These Ljj(m,n) indices seem to be good sources of the susceptibilitv for branehing derived BIA type branehing indices.
Introduction
A number of topological indices have been developed and tested for their performance as branehing indices or indices of substances' properties. ' Topological indices have been correlated with several phvsical, chemical, and biological properties of molecules. However, even several properties of alkanes stili cannot be well deseribed with particular available indices3 and combinations of them have to be used.4 In spite of that, interest in topological indices has grown remarkablv during recent years.
A substantial part of topological indices is derived from one or another matrix associated with molecular strueture. Estrada5 developed a matrix that enables the derivation of an infmite number of indices. We6 presented some types of matrices that enable the derivation of an infmite number of indices, too, and we have shown that these matrices represent a step in unification of several matrices used to derive topological indices, i.e. of the adjacenev matrix, the distance matrix, the reciprocal distance matrix, etc. The characteristics of some groups of indices derived with help of the generalized vertex-degree, vertex distance matrices have been studied:
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-    the summation derived W(m,n) indices6 which include besides the Wiener index7 W also some Ivanciuc indices,
-    the summation derived "mean degree of vertices" indices,9 which include the Randič indexx,'°
i i                                                                                                                                                 n
-    the summation derived Vij(m,n) indices,    which include besides the Wiener index
& 1 9  1 f\                                                                        10
and several Ivanciuc indices '  "   also the Randič index %,
i n
-    the susceptibilities for branching of the W(m,n) indices,    and the difference derived
1 o
indices. In present paper are studied the Ljj(m,n) indices, which are the largest eigenvalues
of the matrices from which the Vij(m,n) indices    have been derived. One index of this
10 90                                                                                                       71 99
group has a long tradition, namelv X\. ' It is also a member of the indices, ' which are the largest eigenvalues of the matrices from which the W(m,n) indices6 or the "mean degree of vertices" indices9 are derived.
The approach used here is in some aspects similar to the search of novel drugs. Searching for novel drugs by combinatorial chemistrv methods, several hundred compounds are svnthesized and tested in order to find among them one or few compounds, which show some activitv. In present study we tested 225 indices from the infmite farnih/ of the Ljj(m,n) indices. Ali of them were tested for their characteristics in order to see the dependence of their values on the values of exponents m and n, on the structural features of alkanes (to perform also the structural interpretation of indices has
7^ 94
been suggested recently ' ), as well as to discover some of them, which either correlate well with physico-chemical properties or seem to be good candidates for branching indices.
Data Notations
The structures of alkanes are presented in shorthand. The notation of ^-alkanes is deliberately different from the branched ones in order to distinguish them easily, e.g. Hp is «-heptane, Oct is «-octane, etc, whereas 223M5 is 2,2,3-trimethylpentane, 3E2M5 is 3-ethyl-2-methylpentane, etc. Other terms are explained on 2,2-, 2,3- and 2,5-dimethyl hexane (22M6, 23 M6 and 25M6) as examples: Ali of them have eight carbons, i.e. eight vertices in the graph (N = 8), as well as four primary carbons, i.e. four vertices of degree
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one (Np = 4). The two branches (i.e. the number of branches, Nbr = 2) in 22M6 are positioned on a quaternary carbon (q), i.e. on a vertex of degree four, which is placed on the periphery (per) of the main chain of the molecule. The two branches in 23 M6 and 25M6 are positioned on tertiary carbons (t), i.e. on vertices of degree three. In 23M6 the branches are adjacent and those in 25M6 are distant, i.e. the separation betvveen branches (s) in 25M6 is larger than in 23M6. On the other hand, in 23M6 is the branch in position 2 closer to the periphery of the main chain, whereas the branch in position 3 is closer to its centre ictr).
Physicochemical properties
The data for the boiling point (BP), density (d), the critical data Te, Pc, Ve, Zc, oce, 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
9S
acentric factor (©) and the refraetive index (nD) were taken from the CRC Handbook or from Lange's Handbook. The data for the liquid molar volume (Vm), the ratios Te /Pc and Tc/Pc used instead of the van der Waals parameters ao and bo, the ratio BP/Tc (reduced BP), and the molar refraetion (MR) were calculated from data presented in the
97
handbooks. The data for Motor Octane Numbers (MON) was taken from Pogliani   and
78                                                                                                                                            70
Gurman et al.,    those for vapour pressure (logVP) from Goli and Jurs,    and those for
7                                         TA
the entropy (S) and quadratic mean radius (R ) from Ren.
The Ljj(m,n) indices and the source matrix
The Ljj(m,n) indices are the largest eigenvalues of the matrix G(m,n) having the main diagonal elements gu, which are equal to zero, and the nondiagonal elements gij, which are equal to \lmx\/nxdjJa, where v, and v, is the degree of vertex i and j, respectively, and dy is the shortest distance from vertex i to vertexy (in alkanes it is the smallest number of bonds betvveen the carbons in question), cf. ref.6 The label Ljj(m,n) indicates that the degree of vertex v, as well as that of vertex v,- is raised to the exponent m, whereas the distance between them is raised to the exponent n. Tested values of exponents m resp. n are: -oo, -6, -4, -2, -1, -Xli,-Xh, -XU, 0, XU, V3, V2, 1, 2, and 3.
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Methods
The one-spot estimation of the relative contribution of structural features to the value of an index is performed using the susceptibility for the increase of the size of the
11                                                         ^ 1
molecule or the susceptibility for branching. The susceptibility is defined as the normalised difference of the indices' or properties' values, Sa^ = Xb/Xa-l, where Sa:t is the susceptibility, X is an index or a property, subscript a refers to the smaller in size or to the less branched structure and subscript b refers to the larger in size or to the more branched structure, as appropriate. Which structure is more branched is concluded by intuition as presented in ref as well as by the Methane based defmition and the n-Alkane based defmition.    Two groups of susceptibilities are used.
In the susceptibility for the increase of the size of the molecule, derived as Saa+i, the subscript a refers to the structure having the same number and type of branches if not explicitly shown othenvise. In these cases, the two alkanes taken into account differ in carbon number by one. For example, in Sy,8 the digit 8 means any octane having the same number and type of branches as a heptane which is represented by the digit 7. In SHP,oct the data of ^-heptane (Hp) and «-octane (Oct) is used. In S2M6,2M7 the data of 2-methyl hexane (2M6) and 2-methyl heptane (2M7) is used, in S2M63M7 the data of 2-methyl hexane (2M6) and 3-methyl heptane (3M7) is used, etc.
In the susceptibility for branching, the first subscript refers to the «-alkane in question and the second subscript refers to another alkane of the same carbon number. For example, in SHp,2M6? Hp refers to ^-heptane and 2M6 refers to 2-methyl hexane.
The one-spot estimation of the relative contribution of a structural feature to the value of the index when the size of the molecule increases, is estimated as follows. The contribution of the number of branches (b) is estimated by b = S33M533M6 - S3M7,3M6- The contribution of the (central) position of branches (c) is estimated by c = V2(S33M5,33m6 -S22M5,22M6)- The contribution of the separation betvveen branches (s) is estimated by s = S24M5,24M6 - §23M5,23M6- The contribution of the change of the substituent from methyl to ethyl (e) is estimated by e = S3E5,3E6 - S3M6,3M7-
The one-spot estimation of the relative contribution of a structural feature to the value of the index when branching of the molecule increases, is estimated as follows.
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The contribution of the number of branches (b) is estimated by b = Soct,33M6 - Soct,3M6-The contribution of the (central) position of branches (c) is estimated by c = Soct,34M6 -Soct,23M6- The contribution of the separation betvveen branches (s) is estimated by s = Soct,24M6 - Soct,23M6- The contribution of the change of the substituent from methyl to ethyl (e) is estimated by e = Soct,3E2M5 - Soct,23M6- An uppercase form of any of these labels is used to label the structural feature having the highest contribution to the value of the index in question. The results of estimations of the relative contribution of structural features are checked by the sequences of isomers obtained by sorting the values of indices corresponding to the isomers in question.
Results and discussion
The Ljj(m,n) indices of methane are equal to zero since gn = 0 by defmition. The Ljj(m,n) indices of ethane are equal to 1. The values of the index Ljj(-oo,-oo) of other alkanes are equal to zero. The values of the index Lij(-V2,-°°) are equal to 1.
Long tradition has the index X/9'20: X\ = Ljj(0,-oo)
Ljj(m,n) indices which are integers
Among the tested Ljj(m,n) indices there are integers Ljj(-oo,-oo), Ljj(-V2,-oo), Lij(-Qo,0),andLij(0,0).
Degeneration of L^(m,n ) indices
Among the tested indices, considering the structures of ali 38 alkanes from propane to octanes inclusive, the indices Ljj(0,-oo), Lij(-V2,-oo), Lij(-15-°°X Ljj(-4,-oo), Ljj(-6,-oo), Ljj(-oo,-oo), as well as the Ljj(m,0) indices are highly degenerated. Some degeneration is observed also among the indices Ljj(-1,-1), Ljj(-2,-l), Ljj(-oo,-l), LijC-oo,1^), Lij(-oo,l), and Ly(-oo,2). Among octanes are degenerated the indices Ljj(0,-co), Lij(-'/2,-20), Ljj(-l,-oo), Lij(-6,-oo), Lij(-oo,-oo), as well as the Lij(m,0) indices.
Correlation o/Ljj(m,n) indices with Vij(m,n) and L(m,n) indices When the structures of ali 38 alkanes from propane to octanes inclusive are
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considered, the correlation between the Lij(m,n) indices and Vij(m,n) indices, assuming the linear relationship of the type: y = ax + b, is good at n > 0 when -1 < m < 2. It is most varying when m < 0 and n < -1. Ljj(-oo,-oo) = Vij(-co,-co) = 0. If only octanes are considered, the best correlations are observed when m « n. There are several m,n combinations where the correlation is low. The Ljj(m,n) and L(m,n) indices are equal when m = 0 or -oo. When the structures of ali 38 alkanes from propane to octanes inclusive are considered, then the lowest correlation is observed around m = -1 and when m > 1. Among octanes, the situation is more differentiated. Low correlation is observed when 0 > m > -2 and m « -n; in the region where -oo < m < - /2 and -00 < n < -2, as well as in the region where m > V2 and n > -V3.
Dependence ofvalues o/*Ljj(m,n) indices ofn-alkanes on the increase ofsize ofthe
molecule
Table 1 indicates whether the Ljj(m,n)„ indices, i.e. the Ljj(m,n) indices of ^-alkanes increase or decrease with the increasing size ofthe molecule. We can see that in majority of tested cases they increase, except in the region represented roughly by -00 < m < -V2 and -00 < n < 0. Within the latter region the values of Ljj(m,n) indices of ethane are greater than at least those of propane and in several cases even greater than the Ljj(m,n) indices of other higher ^-alkanes. There are also indicated the shapes ofthe dependence ofvalues of Lij(m,n) indices of ^-alkanes on the size ofthe molecule. At m < -V2 or m < -1 and n < 0, the values of Lij(m,n) indices of ^-alkanes decrease on the increase ofthe size of the molecule. There exist also two intermediate domains. In the domain labelled C10, Ch, Pe, Bu, and Pr, on increasing the size ofthe molecule the values of Ljj(m,n) indices of ^-alkanes decrease to a minimum at the specified «-alkane and beyond that minimum the values start to increase; i.e. there exists the combination ofthe decrease of values of Ljj(m,n) indices at lower ^-alkanes and of the increase at higher ones. In the domain labelled * there exists the combination of the shapes of increasing values of n-alkanes. Outside these domains, the values of Ljj(m,n) indices of ^-alkanes increase with the size ofthe molecule, resembling either a logarithmic-like increase, i.e. when n < 0 at m > 0 or at n < 1 at lower m, or an exponential increase in other cases.
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Table 1. The changes of values of Lij(m,n) indices of «-alkanes on increasing carbon number. Tested were the carbon numbers from C2 to Ci5.
0
++
M		
m>0 0 -v4 -v3 -v2 -1 -2 -4 -6 -oo	+        +        +        +        +        +        +        +i++ +           +           +           +           +           +           +           +;/ +           +           +           +           +           +           +           +           + +           +           +           +           +           +           +           +           + — :  +      +      +      +      +      +      +      +	++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++    ++    ++    ++    ++
	-   i C,0 i =+     +     +      +      + -   i Cm    Pe    Bu    Pr i   + o     -------  i —	*   : ++    ++    ++    ++ i
		+ + +;++++ +     +     +  j ++   ++
	-oo     -6      -4      -2      -1     -72    -73    -74     0      74     73     72      1    n>l	
	n	
The values of index are equal to zero, except for ethane
The values of index do not change with the size of the molecule
The value of Ljj(m,n) increases linearlv with the size of the molecule
The value of Ljj(m,n) decreases hyperbolically with the size of the molecule
The value of Ljj(m,n) increases exponentially with the size of the molecule +           The value of Ljj(m,n) increases with the size of the molecule resembling the shape of log(size) or
root(size) *            The combination of shape + at low alkanes and shape ++ at higher ones
Pr (Bu, Pe, Cio, C14):         At the size of the molecule corresponding to propane («-butane, «-pentane,
«-decane, «-tetradecane, resp.) is observed the lovvest Ljj(m,n)„ value; beyond that size the values
are increasing
Lij(m,n)Et = Lij(m,n)Pr Dashed lines indicate the intermediate domains between different types of dependence
The values  of the indices  Ljj(-co,n) of ^-alkanes  can be expressed simply: Lij(-oo,n)„ = {N-Yf, where Af is the number of carbon atoms.
Changes of values o/Lij(m,n) indices due to the increase of size ofthe molecules ofother
alkane isomers How the values of Ljj(m,n) indices of other alkanes change on increasing the size of
the molecule when the main chain of the alkane is elongated preserving the branched
structure, is estimated using the susceptibilitv for the increase ofthe size ofthe molecule
(S7,8) and indicated in Table 2. In most cases the values of Ljj(m,n) indices of ali alkane
isomers increase with increasing size of the molecule, except in some cases when m <
-V2 and n < 0. In the latter čase, there is an intermediate domain, where the values ofthe
Ljj(m,n) indices of some isomers increase with the size ofthe molecule, whereas those of
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other isomers decrease, as well as a domain at low values of exponent m, where the values of Ljj(m,n) indices of ali isomers decrease with the size of the molecule. Also in this respect the Lij(m,n) indices differ from the Vij(m,n) and L(m,n) indices in several cases.
Table 2. The changes of values of L;j(m,n) indices of other alkane isomers on increasing the size of the molecule expressed using the sign of S78.
m
ES
5-4-1-6-
3-6-6-
NS
-oo
-72     -73     -74
n
n>0
1-
NS ES
The values of Ljj(m,n) are equal to zero, except for ethane
The values of Ljj(m,n) do not change with the size of the molecule
The value of Ljj(m,n) increases with the size of the molecule among aH isomers
The value of L;j(m,n) decreases with the size of the molecule among ali isomers
The value of L;j(m,n) of one isomer decreases and that of other isomers increase with the size of
the molecule
The values of index do not depend on the size of the molecule
Ali isomers increase equally; the index measures only the size of the molecule
Dashed lines indicate the transition domains between different types of dependence
How the structural features ofalkanes influence the values o/*Ljj(m,n) indices when the
size ofthe molecule increases
The comparison of values of Lij(m,n) indices when the carbon number ofalkanes in question increases by one, i.e from heptanes to octanes, allows some conclusions about the contribution of particular structural features. This comparison is based on the one-spot estimation of the relative contribution of a structural feature to the value of the index, see Methods.
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Table 3. Structural features of alkanes having the highest influence on values of Lij(m,n) indices when the size of molecules increases.
m														
3 2 1 v2 v3 v4 0 -v4 -v3 -v2 -1 -2 -4 -6 -oo	Se Se Se Se Se Se Se	Se Se Se Se Se Se Se	Se Se Se Se Se Se Se	Se Se Se Se Se Se Se Se Se	Sb Sb Sb Sb Sb Sb		Bs	B B B B B B NB -B -B -B -B -B -B -B NS	B-s B-s	-Sb -Sb -Sb -Sb -Sb -Sb			-Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb -Sb	-Sb
							Sb Sb Sb Sb Sb							-Sb
					Sb	Sb Sb Sb Sb			-Sb	-Sb -Sb -Sb				-Sb
					Se Se Se Se				-Sb -Sb -Sb			-Sb		-Sb
						Se | Sb Se | Sb			-Sb -Sb -Sb -Sb		-Sb	-S-e		-Sb
											-S-e	-S-e		-Sb
						Se  Se	Se		-S-e -S-e		-S-e -S-e			-Sb
	S-b	S-b	S-b		Se Se	S-b S-b	-Bs -Bs -Bs -Bs -B-s -S-e -S-e -S-e		-S-b -S-b		-S-e	-S-e		-Sb
	-Bs 0 "b7	-Bs -Bs ~Bs"	-Bs -Bs Sb S-b			S-b -Bs			-B-s -B-s!-S-b -B-s -B-s -S-b -B-s -B-s -B-s -Bs i -Bc i -B-s			-S-e -S-e -S-e -S-e -S-e -S-e		-Sb
				-Bs -Bs -Bc	-Bs -Bs -Bs -Bs -Bs -Bs -B-s -B-s -B-s -S-e -S-e -S-e -S-e -S-e -S-e -S-e -S-e -S-e									-Sb -Sb
	Sb Sb Sb 0	Sb S-b												-Sb
			-Bc	-S-e -S-e -S-e					Se  Se Se  Se Se  Se		-Ec			-Sb
		-Bc	-S-e								Ec Ec			-Sb
		-E-s	-E-s									Sb		-Sb
	-oo	-6	-4	-2	-1  -V2 -73		-v4	0	V4  V3  72  1				2	3
	n													
B or b    - the influence of the number of branches
c            - the influence of central position compared to the peripheral position of branches
E or e     - the influence of ethyl vs. methyl group
S or s     - the influence of separation between branches
The most influential structural feature is presented with the uppercase letter
- sign: The increase of that structural feature causes a decrease in the value of the index
Meaning of combinations of labels:
-Bc: -b>c>..          Bs: b>s>..           B-s: b>-s>..        -Bs: -b>s>..         -B-s: -b>-s>..
Ec: e>c>..           -Ec: -e>c>..         -E-s: -e>-s>..
Sb: s>b>..            S-b: s>-b>..        -Sb: -s>b>..         -S-b: -s>-b>..       Se: s>e>..            -S-e: -s>-e>..
0: The value of index is equal to zero NB: Does not index branching NS: Does not index the size of the molecule
This contribution is superimposed to the contribution of the increase of the size of the molecule observed at ^-alkanes. In Table 3 is presented the situation when the size of the molecule increases from a heptane to an octane by elongation of the main chain retaining the branched structure. There can be seen that depending on the combination of exponents m and n, the highest contribution has either the number of branches (indicated by label B) or the separation between branches (indicated by label S), and in few cases also the type of branches, i.e. if a methyl group is replaced by an ethyl one (indicated by label E). A higher value of the structural feature contributes either to a higher increase of
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the value of the index (no sign before the label) or to a higher decrease of the value of the index (the - sign before the label). At n = 0, only the number of branches and the type of the branched structure (i.e. whether the branch bearing carbon is tertiary or quaternary) influence the value of index.
Regarding the influence of structural features, several Lij(m,n) indices differ from the Vij(m,n) indices11 and this can be the reason for the cases of low correlation betvveen these indices noticed above.
Table 4. Structural features of alkanes having the lowest influence on values of Lij(m,n) indices when the size of the molecules increases. Abbreviations as in Table 3.
m														
3	bc	bc	bc	bc	ec	ec	ec	ec	-e-c	-e-c	-e-c	-e-c	c-e	c-e
2	bc	bc	bc	bc	ec	ec	ec	ec	-e-c	-e-c	-e-c	-e-c	-e-c	-e-c
1	bc	bc	bc	bc	ec	ec	ec	ec	-e-c	-e-c	-e-c	-e-c	-e-c	-e-c
'/2	bc	bc	bc	cb	bc	ec	ec	ec	-e-c	-e-c	-e-c	-e-c	-e-c	-e-c
'/3	cb	cb	cb	cb	cb	bc	ec	ec	-e-c	-e-c	-e-c	b-c	-e-c	-e-c
'/4	cb	cb	cb	cb	cb	bc	ec	ec	-e-c	-e-c	b-c	b-c	-e-c	-e-c
0	c-b	c-b	cb	cb	cb	cb	cb	cb	-cb	-cb	-cb	-cb	-e-c	-e-c
"'/4	ec	ec	ec	c-b	c-b	ec	ec	ec	-e-c	-c-e	-b-c	-c-b	-e-c	-e-c
"'/3	ec	ec	ec	-bc	-bc	ec	ec	ec	-e-c	-e-c	-c-e	-c-b	-e-c	-e-c
"'/2		-ce	-e-c	ec	ec	ec	ec	ec	-e-c	-e-c	-c-e	-c-b	-e-c	-e-c
-1	C	c-e	-e-c	-c-e	ec	-e-c	ec	ec	-e-c	-e-c	-e-c	-c-b	-e-c	-e-c
-2	ce	ce	-c-e	-s-e	-c-e	-c-e	-c-e	-c-e	ce	se	c-e	-c-b	-e-c	-e-c
-4	ec	-c-e	-s-e	-c-b	-cb	-c-b	-c-b	-c-b	c-b	c-b	sb	-cb	-e-c	-e-c
-6	ec	-s-e	-c-e	-cb	-cb	-cb	-cb	-cb	c-b	c-b	sb	b-c	-e-c	-e-c
-oo		b-c	-cb	-cb	-cb	-c-b	-cb	-cb	c-b	c-b	sb	-e-c	-e-c	-e-c
	-oo	-6	-4	-2	-1	-v2	-v3	-V4  0  V4		v3	v2	1	2	3
	n													
Meaning of combinations of labels:
bc: ..>b>c cb: ..>c>b -ce: ..>-c>e ec: ..>e>c sb: ..>s>b
b-c: ..>b>-c c-b: ..>c>-b -c-e: ..>-c>-e -e-c: ..>-e>-c se: ..>s>e
-bc: ..>-b>c -c-b: ..>-c>-b
-s-e: ..>-s>-e
-b-c: ..>-b>-c -cb: ..>-c>b
ce: ..>c>e
c-e: ..>c>-e
In Table 4 are presented for comparison the structural features that have a lower contribution than those presented in Table 3. hi Table 4 can be seen a higher variabilitv with exponents m and n than in Table 3.
A. Perdih, B. Perdih: Some Topological Indices Derivedfrom the vmct Matrix. Part 8. Lij(m,n) Indices
Acta Chim. Slov. 2003, 50, 161-184.
171
Dependence o/*Ljj(m,n) indices on branching Increase or decrease with branching
Whether the values of the Ljj(m,n) indices increase or decrease with branching is presented for octanes in Table 5.
Table 5. Schematic presentation of the change of values of Lij(m,n) indices of octanes on increasing branching.
m											
3	+	+	+	+	+	+	+	+	+	6+	3+
2	+	+	+	+	+	+	+	+	11+	3+	-
1	+	+	+	+	+	6+	2+	-	-	-	-
0<m<l	+	+	+	+	i    -	-	-	-	-	-	-
0 -v4	+ +	+ +	+ +	+ +	i NB	-	-	-	-	-	-
					+	+	i 12+	1+			-
-v3 -v2 -1	+ 0	+ +	+ +	+ + +	+ + +	+ + +	+ + +	10+	-	-	-
								+ +	1+ +		-
	-	-	1+ i								-
-2 -4		-	4+  ;	+ +	+ +	+ +	+ +	+ +	+ +	1+ 2+	-
		5+	+								-
-6	-	+	+	+	+	+	+	+	+	2+	-
-oo	0	+	+	+	+	+	+	+	+	2+	-
	-oo	-6	-4	-2 < n < 0	0	v4	v3	v2	1	2	3
	n										
The value of the Ljj(m,n) indices of aH isomers increases on increasing branching
The value of the Ljj(m,n) indices of aH isomers decreases on increasing branching
The value of the Ljj(m,n) index of one isomer is higher than the value of the Lij(m,n) index of
«-octane, vvhereas the values of Ljj(m,n) indices of other isomers are lovver
The value of the L;j(m,n) index is equal to zero
NB   : The Ljj(m,n) index does not index branching
Dashed lines indicate the transition domains between different types of dependence
0
The situation in Table 5 is similar to, but not equal to that observed among the
77
L(m,n) indices. hi both cases we can observe in the plane of exponents m and n two domains where the values of indices decrease with increasing branching. They are separated from the domain, where the values of indices increase with increasing branching, by intermediate domains. In the intermediate domains, the values of indices either do not depend on branching or the values of indices of some isomers increase with branching whereas the values of indices of other isomers decrease with branching. The
77
different positions of the intermediate domains among the L(m,n) indices    and Vij(m,n)
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Acta Chim. Slov. 2003, 50, 161-184.
indices    compared to those among the Ljj(m,n) indices seem to be responsible for low correlations betvveen them, which are mentioned above.
Influence of structural features
The comparison of values of the Lij(m,n) indices when branching increases, allows some conclusions about the contribution of particular stractural features. hi Table 6 can be seen that depending on the combination of exponents m and n, the highest contribution to the value of Ljj(m,n) indices due to branching has either the number of branches (indicated by label B) or the presence of ethyl groups (indicated by the label E) or even the separation betvveen branches (indicated by label S).
Table 6. Structural features having the highest contribution to the value of a L;j(m,n) index due to the increase of branching.
m															
3 2 1 v2 v3 v4 0 -v4 -v3 -v2 -1 -2 -4 -6 -oo	B-s B-s B-s B-s B-s B-s B-s B-s B-s	B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s	B-s B-s B-s B-s B-s B-s B-s B-s B-s B-s	B-s B-s B-s Be Be Be Be Be Be Be Be B-s B-c B-c B-c	B-s B-s Be Be Be Be Be Be Be B-s B-s Be B-c B-c B-c	B-s B-s Be Be Be Be Be Be Be Be B-s B-s B-c B-c B-c	B-s B-s Be Be Be Be Be Be Be Be B-s B-s B-s B-s B-s	B-s B-s Be Be Be Be Be Be Be Be B-s B-s B-s B-s B-s	B B B B B B B B B B B B B	Bs Bs	Bs Bs -B-e -B-e -B-e -B-e -B-e	Bs Bs -B-e -B-e -B-e -B-e -B-e -B-e	Sb -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -Es	S-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e -B-e	-E-c
															-B-e
										-E-b					-B-e
										-B-e -B-e -B-e -B-e B-e B-e B-e B-e Bs Bs Bs Bs					-B-e -B-e -B-e -B-e
											-E-c				-B-e
											B-e B-e Bs Bs Bs Bs Bs	-E-c			-B-e
												B-e Bs Bs Bs Bs Bs			-B-e
	-S-c -S-c -S-c -S-c	-S-b; -Se													-B-e
		-S-c	-Sb										Bs Bs Bs Bs		-B-e
		-S-e B-c B-c	B-c B-c B-c												-B-e -B-e -B-e
	-oo	-6	-4	-2	-1	-v2	-v3	-v4	0	v4	v3	v2	1	2	3
	n														
Labels: The label has four elements, e.g. b>c>e>s. In Table 6 are presented the former two in the form, e.g. Bc meaning b>c>.., whereas in Table 7 are presented the latter two, e.g. in the form es meaning ..>e>s Meaning of combinations of labels:
B-c: b>-c>.. -E-b: -e>-b>. Sb: s>b>.. -S-e: -s>-e>..
Be: b>e>.. -E-c: -e>-c>. -Sb: -s>b>..
B-e: b>-e>.. -Es: -e>s>.. -S-b: -s>-b>..
-B-e: -b>-e>..      Bs: b>s>..            B-s: b>-s>..
-S-c: -s>-c>..       S-e: s>-e>..          -Se: -s>e>..
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The contribution of ethyl groups or of the separation between branches is higher than that of the number of branches mainly among the Lij(m,n) indices positioned in the intermediate domains observed in Table 5. A higher value of the structural feature contributes either to a higher increase of the value of the index (no sign before the label) or to a higher decrease of the value of the index (the - sign before the label). At n = 0, only the number of branches and the type of the branched structure (i.e. whether the branch bearing carbon is tertiary or quaternary) influence the value of the Ljj(m,n) index. Table 7 presents structural features that have a lower contribution than those presented in Table 6. Indices Ljj(-oo,-oo) and Lij(-1/2,-c0) do not index any contribution of structural features.
Table 7. Two structural features having the lowest contribution to the value of a L;j(m,n) index due to the increase of branching.
m														
3	ec	ec	ec	ce	ec	ec	ec	ec	-e-c	-e-c	-e-c	-e-c	-c-b	-bs
2	ec	ec	ec	ec	ec	ec	ec	ec	-e-c	-e-c	-e-c	-c-b	-cs	-cs
1	ec	ec	ec	ec	-se	c-s	c-s	c-s	-cs	-cs	-cs	-cs	-cs	-cs
'/2	ec	ec	ec	-se	c-s	c-s	c-s	c-s	-cs	-cs	-cs	-cs	-cs	-cs
'/3	ec	ec	ec	-se	c-s	c-s	c-s	c-s	-cs	-cs	-cs	-cs	-cs	-cs
'/4	ec	ec	ec	c-s	c-s	c-s	c-s	c-s	-cs	-cs	-cs	-cs	-cs	-cs
0	ec	ec	ec	c-s	c-s	c-s	c-s	c-s	-cs	-cs	-cs	-cs	-cs	-cs
"'/4	ec	ec	ec	-se	c-s	c-s	c-s	c-s	-cs	s-b	-cs	-cs	-cs	-cs
"'/3	ec	ec	ec	-se	-se	c-s	c-s	c-s	-cs	-cs	s-b	-cs	-cs	-cs
"'/2		ec	ec	-se	ec	-se	-se	-se	-cs	-cs	-cs	-cs	-cs	-cs
-1	-b-e	-c-e	-bc	-se	ec	ec	ec	ec	-cs	-e-c	-e-c	-cb	-cs	-cs
-2	-e-b	-e-b	-ce	-ce	-se	-c-e	-c-e	-c-e	ce	ce	c-e	-e-c	-cs	-cs
-4	-e-b	-cb	-se	-se	-se	-se	-c-e	-c-e	ce	ce	ce	-e-c	-cs	-cs
-6	-e-b	-se	-se	-se	-se	-se	-c-e	-c-e	ce	ce	ce	-e-c	-cs	-cs
-oo		-se	-se	-se	-se	-se	-c-e	-c-e	ce	ce	ce	-e-c	-cs	-cs
	-oo	-6	-4	-2	-1	-v2	-'h	-V4  0	v4	v3	v2	1	2	3
								n						
Meaning of combinations of labels:
-bc: ..>.b>c         -bs: ..>-b>s          -b-e: ..>-b>-e
-cb: ..>-c>b         -c-b: ..>-c>-b       ce: ..>c>e            -ce: ..>-c>e          -c-e: ..>-c>-e       c-s: ..>c>-s
-cs: ..>-c>s                                    ec: ..>e>c            -e-c: ..>-e>-c
s-b: ..>s>-b          -se: ..>-s>c          -se: ..>-s>e
Size ofthe molecule
Index Lij(0,0) indexes only the size ofthe molecule: Ly(0,0) = N-l, where Af is the
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174
Acta Chim. Slov. 2003, 50, 161-184.
number of carbon atoms. Other Ljj(m,n) indices except Ljj(-co,0) index the size of the molecule and also the influence of other structural features. In majority of cases, except in some cases when m < -xl2 and n < 0, cf Table 2, the contribution of the size of the molecule to the value of a Ljj(m,n) index is greater than the contribution of structural features indicating branching. This fact will not be mentioned again below unless considered necessary.
Number ofbranch.es
The index Ljj(-co,0) indexes only the number of branches. The contribution of the number of branches to the value of other Ljj(m,n) indices is in most cases the major one, Table 6.
Type ofthe branched structure
Ljj(m,0) indices, which are not mentioned above in the text, index the size of the molecule, the number of branches, as well as the type of the branched structure, i.e. whether the branch bearing carbon is tertiarv or quaternary. They indicate that the structure having a quaternary carbon is more branched than that having two tertiary carbons.
Type of branches
The label E in Table 6 indicates that at some combinations of exponents m and n, in the Ljj(m,n) indices the exchange of a methvl group for an ethyl group in the structure of an octane contributes to the value of index more than any other structural feature, whereas in several cases (label e) it is the second greatest contribution.
Position of branches
To the value of a Lij(m,n) index the position of branches in most cases contributes less than the number of branches and the type of branches.
Separation between branches
The separation betvveen branches contributes to the value of indices Lj,(3,l), Lj,(3,2), Ljj(2,l), Lij(-6 to-1,-co), Ljj(-4 to -1,-6), Ly(-2,-4), and Ly(-l,-4) more than any other structural feature contributing to branching. It has the second greatest contribution to the values of several indices in the are ofthe m,n plane extending from (-co,l) over (0,-co) to
(3,V2).
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175
Correlation ofphysicochemicalproperties with Lij(m,n) indices
The values of tested Ljj(m,n) indices were correlated with values of 23 or 24 physicochemical properties, as applicable. Best correlations in a linear relationship were sought for in order to have a relationship, which is as simple as possible. At this step, the intention is not to fmd the best possible QSPR models, but the domains of exponents m and n, where the best linear correlation can be expected. For this reason, the results are presented in the plane of exponents m and n; in Fig. 1 for data of 38 alkanes from propane to ali octanes inclusive and in Fig. 3 for data of octanes. Where the data of 2,2,3,3-tetramethyl butane is lacking (i.e. in MON and logVP), this isomer is not included.
Figure 1. Positions of 23 physicochemical properties determined by the highest observed correlation coefficient r (in parentheses, see below) with Lij(m,n) indices for data of alkanes from propane to octanes inclusive, in the plane of exponents m and n.
m															
3															
2															
1						*									MON
v2			de												
v3								Te	BP	VP	Hv		Zc		
v4										**			B/T		
0	(*l)								M	Vm	Ve	T/P	(DO		
-v4														ffl	
-v3									AHP						
-v2															
-1															
-2															
-4			A												
-6															
-oo															
	-oo	-6	-4	-2	-1	-v2	-v3	-v4	0	v4	v3	v2	1	2	3
	n														
Labels, (r): MON (-0.803), de (0.855), Te (0.985), Zc (-0.699), Vm (0.980), Ve (0.991), o (0.973), AHf°g
(0.992), A (0.674)
*:nD (0.895), d (0.931)     BP: BP (0.991), B (0.975)              VP: logVP (-0.986)
Hv: AHv (0.990)               **: Tc2/Pc (0.9995), Pc (-0.960)     B/T: BP/Tc (0.980)
M: Mw (1), MR (0.998)    T/P: Tc/Pc (0.998)                         (D): C (-0.961), ac (0.948)
(ki): largest eigenvalue of the A(djacency matrix)19'20 (Di): largest eigenvalue of the D(istance matrix)
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The pattern of Fig. 1 here is, on the one hand, different from that of Fig. 1 in ref. relating to the Vij(m,n) indices. The reason is obvious; the pattern reflects in both cases the vicinity of indices having a linear or nearly linear increase with carbon number.
Here, the pattern in Fig. 1 is, on the other hand, quite different from that in Fig. 3. In Fig. 1, there is a functional relationship betvveen the molecular weight (Mw) and the index Ljj(0,0), which indicates only the size of the molecule and increases linearly with carbon number. The majority of other tested physicochemical properties correlate best with other indices, derived using exponents m and n not very different from zero. They indicate that a nearly linear increase with carbon number is an important characteristic of an index to index the dependence of many physicochemical properties on the size of the molecule. Most of mentioned indices have a low but positive value of exponent m. The best correlations with data of physicochemical properties of alkanes from propane to octanes inclusive are observed at Mw (r = 1), Tc2/Pc (r = 0.9995), MR (r = 0.998), Tc/Pc (r = 0.998), AHf°g (r = 0.992), Ve (r = 0.991), BP (r = 0.991), and AHv (r = 0.990).
Figure 2. Positions in the plane of exponents m and n of best observed correlation coefficients of phvsicochemical property Tc2/Pc (and the van der Waals parameter ao) and Ljj(m,n) indices for data of alkanes from propane to octanes inclusive.
m															
3															
2															
1									0	n	n	n	¦		
v2						0	0	0	n	•	n	n	0		
v3						0	0	0	n	o	n	n	0		
v4						0	0	0	n	•	o	n	0		
0						0	0	0	0	n	n	•	0		
-v4									0	0	0	n	n		
-v3									0	0	0	0	¦		
-v2											0	0	n		
-1													0	0	
-2														0	
-4														0	
-6														¦	
-oo														0	
	-oo	-6	-4	-2	-1	-v2	-v3	-v4	0	v4	v3	v2	i	2	3
	n														
Open symbols: O: r> 0.999, D: 0.99 < r < 0.999, 0: 0.95 < r < 0.99, no symbol: r < 0.95
Closed symbol: local maximum
A. Perdih, B. Perdih: Some Topological Indices Derived from the vmct Matrix. Part 8. Lij(m,n) Indices
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177
Also the five best correlations of these physicochemical properties with tested Ljj(m,n) indices (not shown) give rise to a similar pattern. The positions of five best correlations of Ve and AHf°g indicate that there is more than one maximum of correlation coefficients in the plane of exponents m and n. This fact is illustrated in Fig. 2 with best observed correlations of Te /Pc with the Lij(m,n) indices.
Figure 3. Positions of 24 physicochemical properties in the plane of exponents m and n, determined by the highest correlation coefficient (data in parentheses, see below) for data of oetanes with L;j(m,n) indices.
m															
3	dVn										*				
2														MR	#   \
1					B/T							Pc			
v2					T/P										ac
v3															
v4										f/p					MON
0	(h)												(Di)		
-v4					®					**					
-v3										Hv					
-v2		C										VP			
-1	Zc			S											
-2													A		
-4															
-6															
-oo												R2			
	-oo	-6	-4	-2	-1	-v2	-v3	-v4	0	v4	v3	v2	1	2	3
	n														
Labels, (r): MR (0.891), Pc (-0.961), ac (0.740), MON (-0.984), ro (-0.994), C (0.967), Zc (-0.729),
S (-0.961), A (-0.775), R2 (-0.927),
dVn: d (0.930), Vm (-0.928), nD (0.923)        *: B (-0.727), Te (-0.848)   #: Ve (0.841), de (-0.824),
B/T: BP/Tc (-0.991)                                    T/P: Tc/Pc (-0.996)           T2/P: Tc2/Pc (0.982)
** BP (-0.852), AHf°g (0.875)                     Hv: AHv (-0.925)              VP: logVP (-0.727)
(A.1): largest eigenvalue of the A(djacency matrix)19'20 (Di): largest eigenvalue of the D(istance matrix)
The pattern of Fig. 3 is different from that of Fig. 1. Evidently, the pattern of Fig. 1 is governed first of ali by the dependence of the values of physicochemical properties on the size of the molecule. The pattern in Fig. 3, on the other hand, is governed only by branehing since the influence of the size of the molecule is excluded. Surprising is the position of several best observed correlations near the diagonals in Fig. 3, characterized
A. Perdih, B. Perdih: Some Topological Indices Derivedfrom the vmct Matrix. Part 8. Lij(m,n) Indices
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Acta Chim. Slov. 2003, 50, 161-184.
by exponents m = n or m = -n. The best correlation is observed at Tc/Pc (r = -0.996),
® (r = -0.994), BP/Tc (r = -0.991), MON (r S (r = -0.961), and Pc (r = -0.961).
-0.985), TcTPc (r = 0.982), C (r = 0.967),
Figure 4. Positions of best five correlations of eight physicochemical properties of octanes with Lij(m,n) indices in the plane of exponents m and n.
m															
3															
2															
1	P5	P3	P4	[32	b5							Pl			
v2				b2	bi				C3	0)4					M4
v3					b3					a3	a4			M5	M2
v4					b4					al	a2	a5			Ml
0	(*0												(Di)		M3
-v4				m2	col										
-v3				0)3	©5										
-v2		Cl	C2	C4	S5										
-1				C5											
-2														P2	
-4															
-6															
-oo															
	-oo	-6	-4	-2	-1	-v2	-V3 : -V4		o	v4	v3	v2	1	2	3
	n														
*PCP
MON
Pc
LhL^nHrCPCP^jfm,!!)))
Ml :Lij(74,3) (-0.98420) M4: L1:(72,3) (-0.98340) P1: Lij( 1,V2) (-0.96136)
P4:Lij(l,-4) (0.94988)
C\:\4rll2,-6) (0.96729)
C4:L1J(-V2,-2) (0.96401)
col: LijC-V^-l) (-0.99425)
©4: L1:(72,74) (0.99211)
bi iLijCVi-l) (-0.99644)
b4:L1J(V4,-l) (-0.99376)
b5:Lij(l,-l) (-0.99106)
P3:Lij(l, -6) (-0.98679)
S             C5:Lij(-l,-2) (-0.96148)
C4:Lij(-1/2,-2) (-0.95617)
Tc2/Pc   al: Lij(74,74) (0.98249) a4:Lij(1/:„1/3) (0.98134)
Tc/Pc
BP/Tc
M2:L1J(V3,3) (-0.98413) M5: Lij(73,2) (-0.98276) P2: Ly(-2,2) (-0.95502) P5:Lij(l,-oo) (0.94978) C2: Lij(-72,-4) (0.96523) C5:Lij(-l,-2) (0.96356) co2: Lij(-74,-2) (-0.99391) co5:L1J(-7:„-l) (-0.99211) b2:L1J(72.-2) (-0.99491) b5:Lij(l,-l) (-0.99258) (32: Li/l,-2) (-0.99077) P5:Lij(l,-oo) (-0.98644) C\:\4rll2,-6) (-0.95777) S5:Li:(-72,-l) (-0.95383) a2:Lij(74,73) (0.98187) a5: Lij(74,72) (0.98036)
M3:Lij(0,3) (-0.98344) P3:Lij(l,-6) (0.94988) C3:L1:(72,0) (-0.96473) co3:Lij(-73,-2) (-0.99270) b3:Lij(73,-l) (-0.99470) P4:Lij(l,-4) (-0.98782) C2: Lij(-72,-4) (-0.95723) a3:L1J(73,74) (0.98185)
* Labelling system:
Physicochemical property (PCP)     label in Fig. 4: Li(m,n) (r (PCP;Lij(m,n)))
(ki): largest eigenvalue of the A(djacency matrix)19'20
(Di): largest eigenvalue of the D(istance matrix)
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The comparison of Fig. 3 here with Fig. 2 in ref. shows that there is a different pattern of positions of best correlation coefficients in the two groups of indices and physicochemical properties in the plane of exponents m and n. The best correlation coefficients are in most cases slightly higher for the Vij(m,n) indices than for the Ljj(m,n) indices. The situation comparing the Vij(m,n) indices11 with Lij(m,n) indices is similar to that when comparing the W(m,n) indices with L(m,n) indices : The largest eigenvalues of the vmdn matrices are mostly slightly better indices of the size of the molecules, whereas the summation-derived indices are mostly slightly better branching indices.
Figure 5. Positions in the plane of exponents m and n of best observed correlation coefficients of the physicochemical property Tc/Pc (and of the van der Waals parameter b0) and L;j(m,n) indices for data of octanes.
m							r<0	<r-		->•	r>0				
3															
2															
1	0	0	0	D	D	0	0	0				0	¦	0	
v2	0	0	n	D	¦	D	0			0	0	0	0	0	
v3	0	0	0	D	D	0				0	0	0	0	0	
v4	0	0	0	D	D	0				0	0	0	0	0	
0	0	0	0	0	0	0	0	¦		¦	0	0	0	0	
-v4				0	0								¦	0	0
-v3													0	0	0
-v2														0	0
-1													¦		0
-2														0	0
-4															0
-6															0
-oo															¦
	-oo	-6	-4	-2	-1	-v2	-v3	-V4	0	V4	V3	v2	1	2	3
	n														
Open symbol: D: 0.99 < abs(r) < 0.999,       0: 0.95 < abs(r) < 0.99,      no symbol: abs(r) < 0.95 Closed symbol: local maximum
In Fig. 4, five positions of best correlations of eight physicochemical properties of octanes correlating best with tested Ljj(m,n) indices are given. They indicate different dependence of best correlation coefficients of physicochemical properties with tested Ljj(m,n) indices on the combination of exponents m and n. The positions of five best
A. Perdih, B. Perdih: Some Topological Indices Derivedfrom the vmct Matrix. Part 8. Lij(m,n) Indices
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Acta Chim. Slov. 2003, 50, 161-184.
correlations corresponding to Pc, C, and © indicate that there is more than one maximum of correlation coefficients in the plane of exponents m and n. The situation is illustrated in Fig. 5 for the best correlating physicochemical property, Tc/Pc. Inspection of ali correlations of tested data shows that the occurence of several local maxima is a general fact.
If we compare the contribution of structural features to the values of Tc/Pc of octanes with the contribution to indices in the highest local correlation maxima in Fig. 5, i.e. to Ljj(V2,-l), Ljj(-'/4,l), and Ljj(-l,l), the sequence of contribution of structural features is -b>-e>-c>s, b>e>c>-s, -b>-e>-c>s, and -e>s>-c>b, respectively. Thus, the indices Ljj(V2,-l) and Ljj(-V4,l) have the same sequence as Tc/Pc, though the former one with opposite sign, which reflects also in the sign of the correlation coefficient. The index Ljj(-l,l), on the other hand, has a quite different sequence of contributions of structural features and in spite of that a quite good correlation. If we compare the relative values of these contributions, Table 8, we can see that they are quite different. It seems that several combinations of these contributions enable a good correlation.
Table 8. Relative contribution of structural features to the value of Tc/Pc and to three indices in local maxima in Fig. 5.
	Tc/Pc	L,i(72,-1)	L,j(-V4,l)	L,i(-l,l)
r (Tc/Pc;index)		-0.9964	0.9917	0.9915
sequence	-b>-e>-c>s	b>e>c>-s	-b>-e>-c>s	-e>s>-c>b
relative     contribution of
b                     -1                     1-11 c                   -0.38                0.28             -0.28            -1.10
s                    0.23                 -0.22             0.14             1.21
e__________-0A6_________038_______-0.40            -1.34
Labels see Table 3.
Ljj(m,n) indices that might be good branching indices.
In Fig. 6, the Ljj(m,n) indices that may be good branching indices are presented in the plane of exponents m and n. The index L;j(-co,0) indexes only the number of branches, i.e. Ly(-oo,0) = Np - 1 = Nbr + 1, where Np is the number of vertices of degree one and Nbr is the number of branches. It is a simple, primitive and degenerated, but a
A. Perdih, B. Perdih: Some Topological Indices Derivedfrom the vmct Matrix. Part 8. Lij(m,n) Indices
Acta Chim. Slov. 2003, 50, 161-184.
181
true branching index presenting the most important infonnation about branching - the number of branches (the number of vertices of degree one).
The indices Lij(m,0), m * -oo and m ^ 0, index the size of the molecule, the number of branches and in addition they indicate that a quaternary structure is more branched than a tertiary one. They are degenerated branching indices, less simple than Ljj(-co,0).
The index Ljj(-oo,l) is interesting since it indexes quite distinctively the size of the molecule, the number of branches, as well as the separation betvveen branches, followed by lower contributions of the type of branches and their position. From the sequence of octanes: Oct < 3Et6 < 4M7 < 3M7 < 2M7 < 3Et3M5 < 33M6 < 22M6 < 3Et2M5 < 34M6 < 23M6 < 24M6 < 25M6 < 233M5 < 223M5 < 234M5 < 224M5 < 2233M4 can be deduced the following rule: n- (Nbr = 0) < It (Nbr = 1; ctr<per) < 2q (Nbr = 2; s = 0; ctr<per) < 2t (Nbr = 2; s = 1; ctr<per) < 2t (Nbr = 2;s = 2)<2t (Nbr = 2; s = 3) < 2qlt (Nbr = 3; s = 1; ctr<per) < 31 (Nbr = 3)<2qlt (Nbr = 3;s = 2)<4q (Nbr = 4), where n- stands for «-alkane, Nbr is the number of branches, t indicates that the branch is positioned on a tertiary carbon, q indicates that the branch is positioned on a quaternary carbon, ctr indicates the position of branch(es) closer to the centre of the main chain of the octane, per indicates the position of branch(es) closer to the end of the main chain of the octane, s is the separation between branches. The relative contribution of structural features is b>s>-e>-c. It indicates that a quaternary structure can be presented as a structure having the separation betvveen branches equal to zero. Thus, whereas the index Ljj(-oo,l) distinguishes clearly the values of the separation betvveen branches, the indices Vij(m,0), n * -oo and n * 0, distinguish only whether the separation betvveen branches equals to zero or it is larger than zero.
There are also 13 Ljj(m,n) indices having a "regular" sequence of isomers: Oct < 2M7 < 3M7 < 4M7 < 3E6 < 25M6 < 24M6 < 23M6 < 34M6 < 3E2M5 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4. They are grouped in a plane characterized by the values of exponents in the vmdn matrix of -1 < m < 0 and -2 < n < 0. It is interesting that although they have the same sequence of isomers, this sequence is caused by three different sequences of contributions of structural features. These Ljj(m,n) indices seem to
17                               ^^
be good sources of the susceptibility for branching derived    BIa type" branching indices.
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Figure 6. L;j(m,n) indices that might be a source of good branching indices of BIA group as well as other indices with interesting characteristics.
m															
3									r<</						
2									t<q						
1									t<q						
'A									t>q						
V3									t>q						
v4									t>q						
0	(*l)								N-\				(DO		
-v4					Ba	Ba	Ba	Ba	Kq						
-v3					Bb	Ba	Ba	Ba	t<q						
-v2	1		Bb    Bc			Bb	Bb	Bb | Kq							
-1							/<</								
-2							1 /<tf 1								
-4									t<q				*		
-6									t<q				*		
-oo	0								Np-\				*		
	-oo	-6	-4	-2	-1	-v2	-v3	-v4	0	V4	v3	v2	1	2	3
	n														
O or 1:    Measures neither size nor branching
Ba, Bb, Bc: The sequence of isomers is Oct < 2M7 < 3M7 < 4M7 < 3E6 < 25M6 < 24M6 < 23M6 < 34M6 < 3E2M5 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4, but the relative contribution of structural features is at Ba: b>e>c>-s , at Bb: b>e>-s>c, and at Bc: b>-s>e>c
t<q:       The index indicates the number of branches as well as the fact that the vertices of degree three contribute less than the vertices of degree four
*           Oct < 3Et6 < 4M7 < 3M7 < 2M7 < 3Et3M5 < 33M6 < 22M6 < 3Et2M5 < 34M6 < 23M6 <
24M6 < 25M6 < 233M5 < 223M5 < 234M5 < 224M5 < 2233M4
Np-1:     Lij(-oo,0) = Np-\=Nbr+\              N-\: 1^(0,0) = N-1
(ki): largest eigenvalue of the A(djacency matrix)19'20
(Di): largest eigenvalue of the D(istance matrix)
Comparing Fig. 3 with Fig. 6 we notice that the indices considered here as potentially good branching indices are not the best indices of tested 24 physicochemical properties of alkanes. This is understandable since none of the 24 tested physicochemical properties of alkanes has a regular sequence of isomers.
Conclusions
The Ljj(m,n) indices are the largest eigenvalues of the generalized type of the vetex-degree, vertex-distance matrix. One index of this group has a long tradition:19,20 Xi = Lij(0,-oo).
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Characteristics of the Ljj(m,n) indices are as follows:
a)   The indices Ljj(-oo,-oo) and Ljj(V2,-°°) are useless.
b)   Index Ljj(0,0) indexes only the size of the molecule: Ljj(0,0) = N-\.
c)   Index Ljj(-c»,0) indexes only the number of branches: Ljj(-co,0) = Np-l = Nbr+l.
d)   Ljj(m,0) indices not mentioned above, index the size of the molecule, the number of branches, as well as the type of the branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary.
e)   Ali other Ljj(m,n) indices index the size of the molecule, the number of branches, the type of branches, the separation betvveen branches, and the position of branches. The type of the branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary is presented by the separation betvveen branches: on quaternary carbons the separation betvveen branches is equal to zero, whereas on tertiary ones it is always greater than zero.
f)    The best correlation of Ljj(m,n) indices with data of physicochemical properties of alkanes from propane to ali octanes inclusive, assuming a linear relationship, is
1       1                          9
observed at the following pairs: Ljj(0,0) and Mw (r = 1), Lj,( /4, /4) and Te /Pc (r = 0.9995), Ljj(0,0) and MR (r = 0.998), Ljj(0,V2) and Tc/Pc (r = 0.998), LjjC-V^O) and AHf°g (r = 0.994), L^O,1^) and Ve (r = 0.991), and ^('/3,0) and BP (r = 0.991). The best correlation with data of physicochemical properties of octanes is observed at the following pairs: Lij(V2,-l) and Tc/Pc (r = -0.996), LjjC-V^-l) and a (r = -0.994), Lij(l,-1) and BP/Tc (r = -0.991), Ly(V4,3) and MON (r = -0.984), L^VJA) and Tc2/Pc (r = 0.982), Lij(-V2,-6) and C (r = 0.967), Ly(-l,-2) and S (r = -0.961), as well as at LijCl,1^) and Pc (r = -0.961).
g)   Thirteen tested Lij(m,n) indices having -1 < m < 0 and -2 < n < 0, have a "regular" sequence of isomers: Oct < 2M7 < 3M7 < 4M7 < 3E6 < 25M6 < 24M6 < 23M6 < 34M6 < 3E2M5 < 22M6 < 33M6 < 3E3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4. This sequence indicates that if the branch(es) is(are) positioned closer to the centre of the main chain of the molecule, then the isomer seems to be more branched than the isomer, which has the branch(es) positioned closer to the periphery of the main chain of the molecule. It is interesting that although these Ljj(m,n) indices have the same sequence of isomers, this sequence is caused by three different sequences of
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contributions of structural features. These Lij(m,n) indices seem to be good sources of the susceptibility for branching derived   BIa type   branching indices.
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Povzetek
Če uporabimo podatke za alkane od propana do vključno oktanov, indeksi L;j(m,n) najbolje korelirajo s fizikokemijskimi lastnostmi v naslednjih primerih: Lij(0,0) in Mw (r = 1), LijCVAt) in Tc2/Pc (r = 0.9995), L1:(0,0) in MR (r = 0.998), Lij(0,72) in Tc/Pc (r = 0.998), Lij(-73,0) in AHPg (r = 0.994), L^O/A) in Ve (r = 0.991), ter Lij(73,0) in BP (r = 0.991). Če uporabimo le podatke za oktane, pa Lij(72,-1) m Tc/Pc (r = -0.996), Ljj(-74,-l) in <n (r = -0.994), Lij(l,-1) in BP/Tc (r = -0.991), L1:(74,3) in MON (r = -0.984), L^A/A) in Tc7Pc (r = 0.982), Lij(-72,-6) in C (r = 0.967), L^-1,-2) in S (r = -0.961), ter 1^(1,72) in Pc (r = -0.961). Trinajst indeksov Lij(m,n), ki imajo -1 < m < 0 in -2 < n < 0, ima regularno zaporedje izomer pri večanju razvejanosti. Videti so dobra podlaga za pripravo indeksov razvejanja vrste BIA.
A. Perdih, B. Perdih: Some Topological Indices Derivedfrom the vmcf Matrix. Part 8. Lij(m,n) Indices