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ON THE ESTIMATION OF PI INDEX OF POLYACENES#
Padmakar V. Khadikar* and Sneha Karmarkar
Research Division, Laxmi Fumigation and Pest Control, Pvt. Ltd., 3, Khatipura, Indore – 452 007, India
Ramendra G. Varma
Department of Chemistry, P.M.B. Gujarati Science College, Indore 452 001, India
#
This paper is dedicated to Professor Ivan Gutman, teacher, inspirer, friend, and proprietor of graph
theory and topology. Received 18-03-2002
Abstract
General formulae are obtained for the PI (Padmakar-Ivan) index of polyacenes. The PI is a newly proposed molecular-graph-distance-based structural descriptor. By means of this result it was possible, for the first time, to examine relative correlation potential of Wiener (W)-, Szeged (Sz)- first-order connectivity (1?), and PI indices in developing Quantitative Structure-Property Relationships (QSPRs) of polyacenes in that hydrophobicity of polyacenes is used as the correlating property.
Introduction
A topological index is a numerical quantity derived in an unambiguous manner
from the structural graph of a molecule.1–5 It is a number extracted by a well defined algorithm from a graphical representation of a molecule. There is good reason to believe that often our difficulties in attributing a meaning to this number lie under deeper chemical theories and higher level languages and not from esoteric approaches to its definition. These indices are graph invariants which usually reflect molecular size, shape, branching, and heterogenicity.4,5
Wiener originally defined6 his index (W) on trees and studied its use for correlations of physicochemical properties of alkanes, alcohols, amines, and their analogous compounds.7,8
The original definition6 of Wiener index (W) was given in terms of edge weights. In an arbitrary tree, every edge is a bridge, that is, after deletion of the edge; the graph is no more connected. The weight of an edge is taken to be the product of the number of vertices in the two connected components. This number also equals the
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number of all shortest paths in the tree, which go through the edge. Therefore, the usual generalization to the Wiener Index (W) on arbitrary graphs is defined to be the same of all distances in a graph.
Another natural generalization was previously put forward by Gutman9,10 and called the Szeged index, abbreviated as Sz. Now, the weights of edges are taken to be the product of the numbers of vertices closed to the two ends of the edge. For reasons to introduce Szeged index, and for its basic properties, uses see reference.11–60
Wiener index (W) is the first, oldest, and widely used topological index. Even to day it is widely used in chemistry.11–60 The Szeged index (Sz) is considered as a modification of Wiener index (W) to cyclic graphs.11 For trees (acyclic graphs) Wiener and Szeged indices coincide. Comparatively little is known on the applications of Szeged index in chemistry.
For the reason of the coincidence of Wiener and Szeged indices in case of trees (acyclic graphs), we have very recently introduced another Szeged / Wiener-like topological index and named it Padmakar-Ivan index, and abbreviated as PI.61–66 Unlike Szeged index (Sz), PI index is different for trees as well as for cyclic graphs, and not much is known about the applicability of PI index in chemistry.
In our earlier publications61–66 we have defined PI index and discussed some of its characteristics and applicability in developing quantitative structure-property-activity (QSPR/QSAR) relationships and observed that PI index is quite useful in this respect. In addition, we have made comparative study of PI index with several other topological indices including W, Sz, and connectivity indices (m?R) and observed that PI index in some cases gives better results.40,67 There we have used physicochemical property and biological activity of n-alkanes, cycloalkanes, alcohols, polychlorinated biphenyls, and monosubstituted nitrobenzene. However, we have not included benzenoid hydrocarbons in such studies. The present communication is, therefore, an extension of such study to benzenoid systems, in that we discuss the methods for the calculation of PI index for polyacenes. The other objective of our present study is to compare W, Sz, and PI indices of polyacenes. Finally, we will make an attempt to use PI index for modeling
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hydrophobicity (logP) of polyacenes, thus attempting QSPR/QSAR for this interesting class of benzenoid hydrocarbons. The results as discussed below show that we are quite successful in this respect.
Results and Discussion Linear Polyacenes
Linear polyacenes (Figure 1) are the most thoroughly investigated homologous series of conjugated molecules (benzenoid systems). A plethora of theoretical work exists devoted to these systems.68–80 However, nothing is known regarding different methods of calculating PI index of polyacenes and their applicability in QSPR/QSAR studies. For QSPR/ QSAR study we need methods for efficient calculations of PI index. Below we present such methods. However, we first repeat in brief, the general definition of PI index.
L1                 1-2                        /-3
12                                    h
Lh Fig. 1. Polyacene molecules
Definition of PI Index
Let G be a molecular graph, the vertex and edge sets of which are represented by V(G) and E(G) respectively. If e is an edge of G, connecting vertices u and v then we write e = uv. The number of vertices of G is denoted by |G|.
The distance between a pair of vertices u, w of G is denoted by d(u,w|G).
We define for e = uv, two quantities ?eu(e|G) and ?ev(e|G). ?eu(e|G) is the number of edges lying closer to the vertex u than the vertex v, and ?ev(e|G) is the number of edges lying closer to the vertex v than the vertex u. Edges equidistant from
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both ends of the edge uv are not counted (taken into account), then the PI index is defined as:61–66
PI =Y   [ Veu (e\G) + r/ev(e\G) ]                                                             (1)
e
The summation goes over all the edges of G
The PI Index of Polyacenes
As stated earlier, in order to be able to undertake studies on the applicability of PI index of polyacenes, we need method(s) of its efficient calculation. In the following sections we present such methods for the calculation of PI index of polyacenes for the first time.
It is well known that polyacenes are linearly para-annelated benzenoids, which possesses transitional symmetry (Figure 1). Chemistry of polyacenes is still very much of interest to synthetic chemists, environmental chemists, cancer research chemists, structural chemists, etc.73–80 We believe that now PI index will be of better value for such diversified studies.
The hexagonal chain whose h hexagons are arranged in a linear manner is denoted by Lh; the respective benzenoid hydrocarbons form the linear homologous series (benzene, naphthalene, anthracene, etc.) Note that the number of hexagons in the
hexagonal chain C is denoted by h. Thus, we have d = Lh C2 = L2, C3 = L3,....., and so
on. The structure of polyacene (Lh) is shown in Fig.1.
From the definition of PI index (equation 1) and from the molecular graph of polyacene molecules (Figures 2 and 3), we observed that for any edge like atbu aibi+1, cidi, diCi+1, i = 1,2,3,… .,n, we have:
cooo
Fig. 2. Molecular graph (Gh) of polyacenes
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ai
a2       a3
ah-1
ah
bi
C1
b2 C2
d!         d2
b3 S3
bh ch
bh+1
Ch+1
d3
dh
Fig. 3. Case of polyacenes used for the estimation of PI index
?(e) = | E] | = 5i - 3
and     ?2(e) = | E2 | = 5h - (5i - 2)
(2)
(3)
Thus giving,
?1(e) + n2(e) = | E1 | + | E2 | = 5h -1                                          (4)
Similarly, for any edge like bici, i = 1,2,3,….., h+1, we have:
?1(e) + n2(e) = | E1 | = | E2 | = 4h                                                       (5)
Hence, PI index of he polyacene will be given by:
PI(Lh) = 4h (5h – 1) + 4h(h +1) = 24 h2                                                    (6)
Recall that Lh has 2 (2h + 1) = n vertices. In view of this PI(Lh) in terms of vertices n is given by the following expression:
PI (Lh) = 3/2 (n-2)2                                                                                (7)
The PI indicates so calculated for the first 20 polycaenes are given in Table 1.
A Case when Calculation of PI Index for Polyacene is Easy
Now consider Figure 4B, in that edges eo, e1, e2, e5 …., er are called vertical edges of the polyacenes and are represented by VE, while those denoted by ei’ and ei” (i = 0,1,2,3,….,n) are called non-vertical edges (NVE) of polyacenes. The sum of the vertical and non-vertical edges is denoted by m, such that VE + NVE = m. The number of hexagons and edges in polyacenes are denoted by h and m respectively.
The lines crossing the vertical and non-vertical edges (Figure 4C) are called elementary cuts and play a distinguish role in the theory of benzenoid system.
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The elementary cuts for the vertical edges (eo, e1, e2,…., er) is presented by Co. The elementary cuts of non-vertical edge for the ith hexagon is represented by Ci’ and Ci” respectively in Figure 4C.
1        2
4A
e/
ef
4B <                q"
Co
Fig. 4. Used for proposing alternative method of estimation PI index of polyacenes
The values of h, n, VE, NVE and m for the first twenty members of the polyacene series are given in Table 1. The PI values estimated from Equation (1) are given in Table 2. The PI values calculated from equations (6) and (7) are also found to be the same.
Now, a critical examination of the calculation PI index using equation (1) indicates that in terms of VE, NVE and m the estimation of PI index is still easy and consists of the following two terms:
(1)  (VE) . (NVE), and
(2)  (m-2) . NVE Thus giving:
PI = (VE) . (NVE) + (m-2) NVE                                                              (8)
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Say for example, for the polyacenes L1 (i.e.,benzene, VE =2,NVE = 4. Therefore, m = VE + NVE = 6.
Therefore, equation (8) gives PI Index for L1 as:
PI (L1) = (2 x 4) + (6 – 2) 4 = 8 + 16 = 24                                                (9)
which comes out to be the same when PI index for L1 is calculated from equations (1), (6) and (7).
Table 1. Calculated values of PI index for polyacenes using equation 1 and the logP values of polyacene molecules.
S. No.	L	n	Names of abbreviated polyacenes	PI	LogP
1	1	6	L 1	24	2.202
2	2	10	L2	96	3.396
3	3	14	L3	216	4.590
4	4	18	L 4	384	5.784
5	5	22	L5	600	6.978
6	6	26	L6	864	8.172
7	7	30	L7	1172	9.366
8	8	34	L8	1536	10.560
9	9	38	L9	1922	11.754
10	10	42	L10	2400	12.948
11	11	46	L11	2904	14.142
12	12	50	L12	3456	15.336
13	13	54	L13	4056	16.530
14	14	58	L14	4704	17.724
15	15	62	L15	5400	18.918
16	16	66	L16	6144	20.112
17	17	70	L17	6936	21.306
18	18	74	L18	7776	22.500
19	19	78	L19	8664	23.694
20	20	82	L20	9600	24.880
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Table 2.   Data needed for alternative method for the calculation of PI index of polyacenes
Polyacenes	L	n	NVE	VE	NVE+VE	PI from eqs. 8, 13 & 14
L 1	1	6	4	2	6	24
L2	2	10	8	3	11	96
L3	3	14	12	4	16	216
L 4	4	18	16	5	21	384
L5	5	22	20	6	26	600
L6	6	26	24	7	31	864
L7	7	30	28	8	36	1176
L8	8	34	32	9	41	1536
L9	9	38	36	10	46	1949
L10	10	42	40	11	51	2400
L11	11	46	42	12	56	2904
L12	12	50	48	13	61	3456
L13	13	54	52	14	66	4056
L14	14	58	56	15	71	4704
L15	15	62	60	16	76	5400
L16	16	66	64	17	81	6144
L17	17	70	68	18	86	6936
L18	18	74	72	19	91	7776
L19	19	78	76	20	96	8664
L20	20	82	80	21	101	9600
Similarly, we can calculate PI indices for the set of 20 polyacene molecules from their respective values of VE, NEV and again found to be the same as presented in Table 1 (Table 2).
A Case when Estimation of PI is Still Easier
A perusal of Table 2 shows that:
VE = h + 1                                                                                       (10)
and     NVE = 4h                                                                                          (11)
So that,
m = 4L + h + 1 = 5h + 1
(12)
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Substituting the values of VE, NEV, and m in equation (8) we get the following:
PI = (VE) (NVE) + (m – 2) NVE                                                       (13)
=    4h (h + 1) + { (5h + 1 – 2) 4h } = 4h (h + 1) + 4L (5h – 1) = 4h { h + 1 + 5h – 1 } = 4h (6h) = 24h2                                                                                      (14)
Thus, the estimation of PI index of polyacenes is made still easier by equation (13).
Comparison of W, Sz, and PI Indices of Polyacenes
It will be interesting to study relatedness among W, Sz, and PI indices of polyacenes. Such a study will provide us a way to predict their relative correlation potential in developing QSPR/QSAR relationships.
As stated above the expression for estimating PI index (equation 6) is:
PI (Lh) = 24 h2 The corresponding expressions for estimating W and Sz are as under:77,78
W (Lh) = 1/3 (16h3 + 36h2 + 26h + 3)                                                 (15)
Sz (Lh) = 1/3 (44h3 + 72h2 + 43h + 3)                                                 (16)
From the aforementioned equations (15) and (16) we observed that Sz index for linear polyacenes have a similar cubic polynomial dependence on molecular size as that of W index, but PI has not. Furthermore, the coefficient involved in these equations (15) and (16) are much higher than in the equation (6). In fact the coefficients of h terms are largest for equation (16). This indicates that the numerical value of Sz will be the largest among the three topological indices under present investigation. The order of the magnitudes of W, Sz, and PI indices follow the following sequence:
Sz (Lh) > W (Lh) > PI (Lh)                                                                   (17)
The values of Sz, W and PI indices for the first 20-polyacene molecules are presented in Table 3. The best account of the relatedness of W, Sz and PI of polyacenes
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could be made by subjecting the data in Table 3 to regression analysis. First step in such a study is to obtain correlation matrix. Such a matrix obtained in the present study is presented in Table 4. The regression analysis gives regression parameters as well as quality of relatedness.81–85 The quality parameters are presented in Table 4.
Table 3.  W, Sz, PI and 1? indices of polyacenes
Polyacenes	W	Sz	PI	1 ?
L 1	27	54	24	3.000
L2	109	243	96	4.967
L3	279	640	216	6.933
L 4	569	1381	384	8.899
L5	1011	2506	600	10.866
L6	1637	4119	864	12.832
L7	2479	6308	1176	14.798
L8	3569	9161	1536	16.765
L9	4939	12766	1944	18.731
L10	6621	17211	2400	20.697
L11	86457	22584	2904	21.663
L12	11049	28933	3456	24.630
L13	13859	36466	4056	26.596
L14	17109	45151	4704	28.562
L15	20831	55116	5400	30.529
L16	25057	66449	6144	32.495
L17	29819	292230	6936	34.641
L18	35149	93571	7770	36.428
L19	41079	109536	8664	38.394
L20	47641	127221	9600	40.360
Table 4.   Correlation matrix for investigating relatedness among W, Sz, and PI indices of poylacenes
	W	Sz	PI
W	1.0000		
Sz	0.9999	1.0000	
PI	0.9887	0.9885	1.0000
Proposed regression expressions:
PI (Lh) = 0.1672 (+ 0.0053) W (Lh) + 1075.3313
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PI (Lt) = 0.0625 (+ 0.0020) Sz (Lt) + 1096.6398
Quality of aforementioned correlations:
Correlation	Se	R	F	P	Q
PI – W	724.1565	0.9887	1001.679	0.000E+00	0.0014
PI – Sz	731.8385	0.9885	980.280	0.000E+00	0.0014
A perusal of Table 4 shows that W, Sz and PI are highly linearly correlated; Sz correlates with W slightly more than PI index; while the correlation of PI index with W and Sz is similar. The corresponding relatedness of PI with W and Sz is given by the following regression expressions:
PI (Lh) = 0.1672 W (Lh) + 1075.3313                                                 (18)
and     PI (Lh) = 0.0625 Sz (Lh) + 1096.6398                                                (19)
The corresponding qualities viz., standard error of estimation (Se), correlation coefficient (R), F-ratio, and probability values also indicates similar relatedness between PI-W and PI-Sz (Table 4).
It is worth recording that in addition to high relatedness, a particular index will be preferred over the other index in that fewer efforts are made for its estimation. Therefore, compared to both W and Sz, PI index is better as it is given by 4h2 only. Thus, least efforts are required for its calculation.
From the aforementioned results and discussion we can conclude that correlation potential of W, Sz, and PI indices are similar. Also that, like W and Sz, PI index also takes care of size, shape and branching. The size-shape dependence of W and Sz is well established[86-88]. But, as discussed below we observe that PI index contains hither to unknown structural features, which dominates over size and shape. This makes PI index to give better results than W and Sz in some cases.61
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Modeling of Hydrophobicity of Polyacenes Using W, Sz, and PI Indices
It is well established that hydrophobic interaction in biosystems can be modelled by logP i.e. logarithm of partition coefficient (P) between octanol-water or in some other suitable organic solvent / water partitioning systems.81–92 Modeling of logP with molecular descriptors (topological indices) has two-fold advantage, first logP is considered as a physicochemical property and secondly it can also be used to represent physiological activity of organic compound acting as drugs.
The logP values89–92 for the first 20 polyacene molecules are presented in Table 5. In addition to W, Sz and PI indices we have also chosen 1?-index67 because it is most widely used topological index in QSPR and QSAR studies. That is now, we are considering relative ability of W, Sz, PI, and 1? indices for modeling, monitoring, estimating logP of polyacenes. These 1? values are presented in Table 3.
Table 5.   Correlation matrix for investigation relatedness among W, Sz, PI, and 1? and their correlation with logP of polyacene molecule
	LogP	W	Sz	1?	PI
LogP	1.0000				
W	0.9264	1.0000			
Sz	0.9258	0.9999	1.0000		
1?	0.9999	0.9273	0.9267	1.0000	
PI	0.9707	0.9887	0.9885	0.9713	1.0000
Proposed regression expressions for modeling logP of polyacene molecules using W, Sz, 1?, and PI indices.
logP = 2.9087 x 10-4 (+ 2.4652 x 10-5) W + 9.171; logP = 1.088 x 10-4 (+ 9.2641 x 10-6) Sz + 9.211; logP = 0.6070 (+ 1.7450 x 10-3) 1? + 0.1240; logP = 1.8027 x 10-3 (+ 9.3062 x 10-5) PI + 6.9680.
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Quality of aforementioned correlations:
Correlating parameter	Se	R	F	P	Q
w	3.3800	0.9260	139.215	3.099 x 10-11	0.2740
Sz	3.3940	0.9260	137.867	3.406 x 10-11	0.2728
1 ?	0.1240	1.0000	120842.000	0.00E+0	8.0645
PI	2.1570	0.9710	375.224	0.000E+0	0.4502
We have first obtained correlation matrix for the parameters W, Sz, PI, 1? and logP (Table 5). This correlation matrix indicates that compared to both W and Sz, PI-index is a better topological index for modeling logP of polyacene molecules. However, 1? is the most appropriate index for this purpose. That is, correlation potential of PI index in modeling logP is inferior to 1? but superior than W and Sz. Whether this is the case with other properties / activities is a problem for further investigation. Attempts in this direction are under way and the results will be published soon.
In order to obtain better insight into the problem of modeling logP with W, Sz, PI, 1? we have subjected the data to regression analysis and obtained corresponding regression parameters i.e., the values of Se, R, F, P and Q (Table 5). Here, Q is the quality factor93,95 obtained from the ratio of correlation coefficient (R), and standard error of estimation (Se) (Q = R/Se). The advantage of using Q is that it takes accounts of R and Se simultaneously. The quality factor Q is directly proportional to R and inversely proportional to Se. Hence, larger the value of R, smaller the Se, higher will be Q, and the better will be the proposed correlation.
The data presented in Table 5 indicates that correlation potential of W and Sz in modeling logP of polyacene molecules is similar. The PI index, is slightly more efficient than W and Sz for this purpose and that 1? is the most appropriate index among W, Sz, PI and 1? for modeling logP. Q-values show that 1? is almost two-fold better index than PI, and that the relative potential of the topological indices in modeling logP follow the following sequence.
1?(Lh) > PI(Lh) > W(Lh) ? Sz(Lh)                                                  (20)
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Once again we state that shape-size dependence of W and Sz is well established. Inspite of high collinearity among W, Sz, and PI, the PI index is found better than both W and Sz. The Q-value of 0.4502 compared to ? 0.2746 or 0.2728 indicates that in addition to size and shape, the PI index contain some other hither to unknown structural feature, which dominates over shape and size. As stated earlier the study in this direction is underway and results will be published soon.
In order to investigate probable unknown parameter(s) hidden in PI index we considered the polynomials for logP = f(L) and logP = f(1?). From the data presented in Table 5 we have:
logP = 1.8022 x 10-3 (+ 9.3062 x 10-5) PI + 6.9680                         (21)
and,    logP = 0.6070 (+ 1.7450 x 10-3) 1? + 0.1240                                  (22)
The first order connectivity index (1?) conveys more information about the number of atoms in a molecule. Therefore, according to equation (22) it is the atomic contribution, which accounts for the exhibition of logP. Similarly, according to equation (21), the PI index is directly related to logP. While the defination of PI index (= 24 h2, equation 6) shows that PI is directly related to number of cycles (h) present in the polyacene molecule. Hence, the unknown parameter hidden in PI index is the cyclicity. That is, in additive to first, shape, branching, the PI index also accounts for cyclicity, thus, hither to unknown structural feature contain in PI index is the cyclicity which domination over shape and size.
Conclusions
The aforementioned results and discussion lead us to the following conclusions:
(1)    The estimation of PI index for polyacene molecules is much simpler than the estimation of W and Sz;
(2)    Both W and Sz are cubic polynomials while PI index is not;
(3)    Coefficient of h parameters in the corresponding expressions proposes the following order of magnitudes of W, Sz, and PI. Sz(Lh) > W(Lh) > PI(Lh);
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(4)  High collinearity among W, Sz, and PI indicates their similar correlation potential in proposing QSPR/QSAR models, and also their similar dependence on shape and size;
(5)    High quality of correlation of logP with PI, compared to both W and Sz, indicates that in addition to size and shape, the PI index depends upon some hither to unknown structural feature which dominates over shape and size.
Experimental
For the calculation of PI index the molecular structures are transformed into their
molecular graphs in that atoms (vertices) are depicted by dots and bonds (edges) by small lines.
Acknowledgements
Authors are highly thankful to Prof. Ivan Gutman for introducing them to this
fascinating field viz., Chemical Graph Theory and Topology, and to Prof. Istavan Lukovits for providing software.
References and Notes
1.    J. Devillers, A.T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breace Science Publisher, Amsterdam, 2000.
2.    L.B. Kier, L.H. Hall, Molecular Structure Description, Academic Press, New York, 1999.
3.    R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, New York, 2000.
4.    J. Devillers, Comparative QSAR, Taylor and Francis, Philadelphia, 1998.
5.    N. Trinajstic, Chemical Graph Theory, 2nd revised ed., CRC Press, Boca Raton, FL, 1992.
6.    H. Wiener, J. Am. Chem. Soc. 1947, 69, 17-20.
7.    I. Gutman, Y.N. Yeh, S.L. Lee, Y.L. Luo, Indian J. Chem. 1999, 32A, 651-661.
8.    There exists a large numbers of papers, reviews, and books dealing with the Wiener Number and its applications: We have mentioned only the aforementioned papers and reviews, up to 1996. References to earlier work can be found in references 2-4.
9.    I. Gutman, Graph Theory Notes, New York, 1999, 27, 9-15.
10.  P.V. Khadikar, N.V. Deshpande, P.P. Kale, A. Dobrynin, I. Gutman, G. Domotor. J. Chem. Inf. Comput. Sci. 1995, 35, 547-550.
11.  P.V. Khadikar, P.P. Kale, N.V. Deshpande, S. Karmarkar, V.K. Agrawal. Commun. Math. Comput. Chem. (MATCH) 2001, 43, 7-15.
12.  A. Dobrynin, I. Gutman, Publ. Inst. Math. (Beograd), 1994, 56, 18-22.
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Povzetek
Prikazane so splošne formule za PI (Padmakar-Ivan) indekse poliacenov. PI je na novo predlagan strukturni descriptor molekularne strukturne formule in razdalj. Na osnovi prikazanih rezultatov je možno primerjati uporabnost Wienerjevih (W)-, Szegedovih (Sz)-povezav prve vrste (1?) in PI indeksov v razvijanju QSPR poliacenov, pri θemer kot korelacijsko lastnost uporabimo hidrofobnost.
P. V. Khadikar, S. Karmarkar, R G. Varma: On the estimation of PI index of polyacenes