The Wiener index W is the oldest molecular-graph-based structure-descriptor. It is defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph. This seemingly very simple topological index could be "upgraded" (a) by using as the distance the sum of the bond lengths along the shortest path, or (b) by using the Euclidean distance between the respective pairs of atoms. Each of these "upgraded" Wiener indices could be computed either (α) for the hydrogen-depleted or (β) for the hydrogen-filled molecular graph. We provide examples showing that none of the modifications (aα), (aβ), (bα), (bβ) yields better results than the ordinary Wiener index, and that there is a very good linear correlation between W and its "upgraded" variants.Centro de Química Inorgánica (CEQUINOR

Castro, Eduardo Alberto

Gutman, Ivan

Marino, Damián José Gabriel

Peruzzo, Pablo José

Journal of the Serbian Chemical Society

English

El Servicio de Difusión de la Creación Intelectual

J. Serb. Chem. Soc. 67 (10)647–651(2002) UDC 54-12+539.2+512.546.1
JSCS-2986 Original scientific paper
Upgrading the Wiener index
EDUARDO A. CASTRO,a IVAN GUTMAN,b# DAMIAN MARINOa and
PABLO PERUZZOa
aCEQUINOR, Departamento de Quimica, Facultad de Ciencias Exactas, Universidad Nacional de La
Plata, C. C. 962, La Plata 1900, Argentina and bFaculty of Science, University of Kragujevac, P. O. Box 60,
YU-34000 Kragujevac, Yugoslavia
(Received 18 June 2002)
Abstract: The Wiener index W is the oldest molecular-graph-based structure-descriptor. It is
defined as the sum of the distances of all pairs of vertices of the molecular graph G, where
the distance is the number of edges in the shortest path connecting the respective vertices,
and where G is the hydrogen-depleted molecular graph. This seemingly very simple topo-
logical index could be "upgraded" (a) by using as the distance the sum of the bond lengths
along the shortest path, or (b) by using the Euclidean distance between the respective pairs of
atoms. Each of these "upgraded" Wiener indices could be computed either () for the hydro-
gen-depleted or () for the hydrogen-filled molecular graph. We provide examples showing
that none of the modifications (a), (a), (b), (b) yields better results than the ordinary
Wiener index, and that there is a very good linear correlation between W and its "upgraded"
variants.
Keywords: Wiener index, topological index, 3D-structure descriptors, chemical graph the-
ory, QSPR, QSAR.
INTRODUCTION
The Wiener index1 (W) is the oldest molecular-graph-based structure-descriptor2 and
its chemical applications3–5 and mathematical properties6,7 are well documented. The
usual explanation as to why W is so successful in QSPR and QSAR studies is based on the
fact that W represents a rough measure of the van der Waals molecular surface area.8 Con-
sequently, in the case of non-polar molecules (such as alkanes3 and cycloalkanes5), W is
proportional to the intermolecular forces,4,9 and is thus related to a number of phy-
sico-chemical properties of the respective compounds (boiling point, heat of evaporation,
heat of formation, chromatographic retention times, surface tension, vapor pressure, parti-
tion coefficients, etc.). The applicability of the Wiener index for the description of
physico-chemical and pharmacologic properties of compounds very different from saturated
hydrocarbons (e.g., fungicidal derivatives of salicylhydroxamic acid,10 antiviral 5-vinyl-
647
# Serbian Chemical Society active member
pyrimidine nucleoside analogs,11 mixed amino acid complexes with copper (II)12,13) moti-
vated us to examine variants of W in which the structural features of such non-hydrocarbon
compounds could be incorporated.
"UPGRADING" THE WIENER INDEX
The original definition of the Wiener index is as follows.1,14 Let G be a molecular
graph14 and u and v two of its vertices. Denote by d(u,v|G) the distance between u and v (=
number of edges in the shortest path connecting u and v). Then the Wiener index is
W = W (G) = d u,v G( )|
u,v

(1)
with the summation going over all pairs of vertices of G. The molecular graph G used in
formula (1) is hydrogen-depleted, i.e., its vertices represent only carbon (and perphaps
some other heavy) atoms, but not hydrogens.
The calculation of W is rather simple, either directly by means of formula (1) or by using
some of the numerous shortcuts, applicable to particular classes of molecular graphs.6,7,14–16
Needless to say, the distances d(u,v|G) in formula (1) are integers, and therefore W is also in-
teger-valued. The obvious drawback of W, as defined via (1), is that it is insensitive to the ex-
istence and position of double and triple bonds as well as of heteroatoms.
Some possible modificiations of formula (1), aimed at bringing the Wiener index
closer to real-world molecules, come readily to mind:
(a) Structure-descriptors Wb and WbH: instead of the graph-theoretical distance d(u,v|G) –
which is just the number of chemical bonds between the atoms u and v – one could use the sum
of the bond lengths along the shortest path, provided the geometry of the molecule is known.
This geometry needs to be computed by means of a suitable molecular-mechanics or molecu-
lar-orbital method. By means of Wb and WbH, single, double and triple bonds will be clearly
distinguished, and their position in the molecule taken into account.
(b) Structure-descriptors We and WeH: Instead of d(u,v|G) one could use the Euclidean
distance between the atoms u and v. Again, the molecular geometry must be known. The
possible advantage of these indices would be that through-space, and not through-bond inter-
actions would be accounted for.
() Usually the molecular graph used for the computation of the Wiener index and its
modifications is the hydrogen-depleted molecular graph, in which vertices represent only
the heavy atoms (carbon and heteroatoms). This choice would result in the indices W, Wb
and We.
() If the Wiener index and its modifications would be calculated on the basis of the
hydrogen-filled molecular graph, then instead of W, Wb and We, one would obtain WH,
WbH and WeH, respectively.
The structure-descriptor We was studied in the past;17,18 it was referred to as the
three-dimensional (or 3D) Wiener index. In the case of acyclic molecules, the quantities W
and WH were shown to be related by means of an exact mathematical relation:19
WH = 9W + (3n + 1)2 (2)
648 CASTRO et al.
where n is the number of carbon atoms. To the authors’ best knowledge, the other, above
mentioned, variants of the Wiener index have so far not been considered in the chemical
literature.
NUMERICAL WORK 1: THE SIX VARIANTS OF THE WIENER INDEX ARE LINEARLY
CORRELATED
The fact that WH and W are linearly correlated is seen from Eq. (2). Therefore the re-
lation between these two structure-descriptors was not examined any further.
In order to test the quality, predictive power and the differences between the remain-
ing five variants of the Wiener index (W, Wb, WbH, We, WeH), two data bases were used: a
set of 39 acyclic monohydroxy alcohols (C1 to C7) from the paper20 and a set of 25 acy-
clic alkenes from the paper.21
The optimized geometries of the respective molecules were determined by means of
the program package Hyper Chem, employing the semiempirical AM1 molecular orbital
method. The quantities Wb, WbH, We, WeH were calculated from the appropriately modi-
fied formula (1), into which bond lengths and interatomic distances were substituted. All
bond length and interatomic distances were in picometers. Consequently, Wb, WbH, We,
WeH are also expressed in pm-units (whereas W and WH are dimensionless quantities).
It was found that the original Wiener index W and its modifications are highly corre-
lated and that these correlations are linear. For alcohols20 the respective regression lines
and their correlation coefficients R read:
Wb = 150.29 W + 35.38 R = 0.9999
We = 124.43 W – 83.02 R = 0.9994
WeH = 1233.3 W + 9168.9 R = 0.9984
whereas for alkenes21 the following values were obtained:
Wb = 156.25 W + 374.83 R = 0.9917
We = 117.3 W + 176.4 R = 0.9966
WeH = 1302.7 W + 7369.8 R = 0.9976
In view of relation (2) and the very good linear correlation between W and Wb, there
was no need to analyze the correlation between W and WbH, which must also be linear and
very strong.
The R-values of the above regressions are all very close to unity, implying that a
strong linear correlation exists between all the six variants of the Wiener index. In other
words: all the "upgraded" versions of the Wiener index bear essentially the same informa-
tion on molecular structrure and none is significantly better than W itself.
UPGRADING THE WIENER INDEX 649
NUMERICAL WORK 2: STRUCTURE-PROPERTY RELATIONS BASED ON THE VARIANTS OF
THE WIENER INDEX
In view of the analysis outlined in the preceding section one must expect that the
QSPR relations based on the modified versions of W will be of similar quality as those
based on W and that "upgrading" the Wiener index will yield little or no gain. This indeed
was found by examining the relation between the various Wiener indices and the boiling
points of alcohols, as well as the molar refractivities of alkenes.
For the linear correlation between the boiling points of alcohols and W, Wb, We and
WeH the correlation coefficients are equal to 0.9367, 0.9375, 0.9349, and 0.9372, respec-
tively. For the linear correlation between the molar refractivity of alkenes and W, Wb, We,
and WeH the correlation coefficients are equal to 0.9779, 0.9812, 0.9652, and 0.9822, re-
spectively. In both cases the increase of accuracy of the correlation when W is exchanged
by Wb, We, or WeH is statistically insignificant and negligible.
CONCLUDING REMARKS
It has been shown that some, seemingly reasonable and theoretically justifiable, mod-
ifications of the Wiener index do not improve at all the applicability and value of this struc-
ture-descriptor in designing quantitative structure-property relations. This may be viewed
as a somewhat surprising and somewhat disappointing finding. From another point of
view, the same finding implies that the original Wiener index – now already half-a-century
old – is a much more valuable topological index than one would expect from its extremely
simple and seemingly naive definition, Eq. (1).
From a practitioner's point of view the dilemma: W or some of its "upgraded" versions
Wb, WbH, We, WeH has now been completely resolved. In QSPR and QSAR studies the
modified Wiener indices will give no better results than the original Wiener index. The cal-
culation of these modified Wiener indices is difficult, requiring the knowledge of molecu-
lar geometry. The calculation of the original Wiener index W is simple. Therefore, unless
some convincing argument in favor of a modified Wiener index is offered, preference
should be given to its original version, Eq. (1).
IZVOD
USAVR[AVAWEVINEROVOGINDEKSA
EDUARDOA.KASTRO,IVANGUTMAN, DAMIANMARINOiPABLOPERUCO
Hemijski odsekFakulteta egzaktnih nauka, Nacionalni UniverzitetLaPlate, LaPlata, Argentina i
Prirodno-matemati~kifakultet uKragujevcu
Vinerov indeksW je najstariji, na molekulskom grafu zasnovani, strukturni deskrip-
tor.Definisan je kao zbir rastojawaizme|u svih parova ~vorovamolekulskog grafaG. Pri
tome, rastojawe je jednakobroju grana koje le`ena najkra}emputu koji povezuje odgovaraju}e
~vorove, a G je "bezvodoni~ni" molekulski graf. Ovaj naizgled veoma jednostavan topo-
lo{kiindeksmogaobi se "usavr{iti" tako{tobi sekaorastojaweuzimalo (a) zbir du`ina
veza du` najkra}eg puta, ili (b) Euklidsko rastojawe izme|u odgovaraju}ih parova atoma.
Svaki od ovako "usavr{enih" Vinerovih indeksa mogao bi se ra~unati na () "bezvodoni-
650 CASTRO et al.
~nom" molekulskom grafu, ili () na "vodoni~nom" molekulskom grafu. Mi navodimo
primerekojipokazuju dani jednaodmodifikacija (a), (a), (b), (b) ne daje boqerezultate
negoobi~niVinerovindeks, kaoidapostoji dobralinearnakorelacija izme|uWiwegovih
poboq{anih varijanti.
(Primqeno 18. juna 2002)
REFERENCES
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6. A. A. Dobryinin, R. Entringer, I. Gutman, Acta Appl. Math. 66 (2001) 211
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12. S. Nikoli}, N. Raos, Crot. Chem. Acta 74 (2001) 621
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16. I. Gutman, J. Serb. Chem. Soc. 58 (1993) 745
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18. M. Randi}, B. Jerman-Bla`i}, N. Trinajsti}, Comput. Chem. 14 (1990) 237
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UPGRADING THE WIENER INDEX 651


Upgrading the Wiener index

The Wiener index W is the oldest molecular-graph-based structure-descriptor. It is defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph. This seemingly very simple topological index could be "upgraded" (a) by using as the distance the sum of the bond lengths along the shortest path, or (b) by using the Euclidean distance between the respective pairs of atoms. Each of these "upgraded" Wiener indices could be computed either (a) for the hydrogen-depleted or (b) for the hydrogen-filled molecular graph. We provide examples showing that none of the modifications (aa), (ab), (ba), (bb) yields better results than the ordinary Wiener index, and that there is a very good linear correlation between W and its "upgraded" variants

PABLO PERUZZO

IVAN GUTMAN

DAMIAN MARINO

EDUARDO A. CASTRO

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