Kailas S. Ahire and Rajashri B. Sawant

Bulletin of the Marathwada Mathematical Society
Vol. 14, No. 2, December 2013, Pages 1–4.
WIENER INDEX AND WIENER POLYNOMIAL IN
ISOMERS OF ORGANIC COMPOUNDS
Kailas S. Ahire,
Arts, Science and Commerce College, Nampur,
Tal. Satana, Dist. Nashik, (M.S.)
And
Rajashri B. Sawant,
S. P. H. Mahila College, Malegaon-camp,
Tal. Malegaon, Dist. Nashik, (M.S.)
Abstract
Graph theory is used to a mathematical model of molecules in order to gain
an insight into the physical properties the chemical compounds. Some physical
properties, such as boiling point, melting point, density etc. are related to the
geometric structure of the compound.
The Wiener Index W is the ﬁrst graph - theoretical invariant used in chemistry
[5]. The Wiener Index is equal to the count of all the shortest distances between all
the pairs of the vertices in a molecular graph. The organic compounds are exclusively composed of carbon and hydrogen atoms. The carbon tree can be represented
as a graph by replacing the carbon atoms with the vertices of the graph. Chemical
bonds are then represented as the edges in the graph. 1
1
INTRODUCTION
The Wiener Index W is a frequently and successfully used structure descriptor in studies,
based on the molecular topology of quantitative relations between the structure and the
activity. Its numerous physical and chemical applications range from prediction of the
boiling points of alkanes [7], to extent of branching of the carbon-atom. Skeleton of an
organic molecule is related to its molecular surface area [8, 1]. Boiling point is a measure
of the forces of attraction between like molecules. Pharmacological applications of W
include the work on cytostatic and antihistaminic activities [4].
A quantitative measure of branching is needful for ﬁnding connections between
molecular structure and physico-chemical properties of chemical compounds. Isomers
are molecules that have the same molecular formula, but have a diﬀerent arrangement
of the atoms in space.
1
@ Marathwada Mathematical Society, Aurangabad, India, ISSN 0976-6049
1
2
2
Kailas Ahire & Rajashri Savant
WIENER INDEX
The Wiener Index is a graph invariant based on the concept of distance in a graph. It is
denoted by W(G) and deﬁned
as the sum of distances between all the pairs of vertices
1 ∑
in G; that is W(G) = 2
dG(u, v). It can be computed by adding the entries in
u,v∈G
upper (or lower) triangular part of the distance matrix of a graph.
We examine the structure including boiling point, density and melting point of
pentane.
C5 H12 has three diﬀerent structural isomers with a linear structure of 5 carbon atoms
iso-pentane and neopentane with a branched structure.
The chemical graph(carbon tree)for pentane is a ﬁve vertex path graph.
The chemical graph(carbon tree)for isopentane is a ﬁve vertex path graph.
The chemical graph(carbon tree)for neopentane is a ﬁve vertex path graph.
The distance matrices of pentane, iso-pentane and neopentane are given below:
WIENER INDEX AND WIENER POLYNOMIAL IN ISOMERS OF . . .
1

1 0
2
1
3
2
4 3
5 4
2 3 4 5
1
0
1
2
3
2
1
0
1
2
3
2
1
0
1

4
3

2

2
0
1

1 0
2
1
3
2
4 3
5 3
2 3 4 5
1
0
1
2
2
2
1
0
1
1
3
2
1
0
2
The Wiener Index of pentane (Figure - 1) is
1 + 1 = 19.

3
2

1

2
0
1

1 0
2
1
3
2
4 2
5 2
3
2 3 4 5
1
0
1
1
1
2
1
0
2
2
2
1
2
0
2

2
1

2

2
0
1+2+3+4+1+2+3+1+
The Wiener Index of iso-pentane (Figure - 2) is
+ 2 = 18.
1+2+3+3+1+2+2+1+1
The Wiener Index of neopentane (Figure - 3) is
+ 2 = 16.
1+2+2+2+1+1+1+2+2
Melting Point (0 C)
Density (g/l)
Boiling Point (0 C)
3
Isomers
pentane iso-pentane
-129.8
-159.9
621
616
36.0
27.7
neopentane
-16.6
586
9.5
MEAN ISOMER DEGENERACY OF WIENER INDEX AND WIENER
POLYNOMIAL
The Mean Isomer Degeneracy of Wiener Index [6, 2] has been deﬁned as µ = Nt ,
where N is the number of distinct structural isomers considered and ‘t’ is the number of
distinct structural values that the Wiener index assumes for these isomers. In an ideal
case, for which each isomer is associated with a distinct W, we have µ = 1. Otherwise
µ > 1 and shows how many isomers share the same value of W. Evidently, the larger is
the isomer degeneracy of W, the smaller is its power to distinguish isomers, a property
essential for many practical purposes.
To illustrate the concept of mean isomer degeneracy, consider the isomeric alkanes with seven carbon atoms. There exists nine C7 H16 isomers (i.e., N = 9) but they
have only seven distinct values of the Wiener index (i.e., t = 7). Consequently, for
the C7 H16 isomers µ = 9/7
The Wiener Polynomial [3, 10] of a connected
∑ = 1.29.
graph G are W(G, q) =
q dG(u,v) where dG(u, v) denotes the distance between
u,v−inG
two vertices u and v in G. The Wiener Polynomial W(G, q) for pentane (Figure-1) is
W (G, q) = q 4 + 2q 3 + 3q 2 + 4q.
The Wiener Polynomial W(G, q) for iso-pentane (Figure-2) is W (G, q) = 3q 3 + 4q 2 + 4q.
The Wiener Polynomial W(G, q) for neopentane (Figure-3) is W (G, q) = 6q 2 + 4q.
4
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Kailas Ahire & Rajashri Savant
HOSOYA INDEX
The Hosoya Index (Z)[9] of a graph G is deﬁned as the number of subsets of the edge
set in which no two edges are adjacent in G. i.e.; Z is the total number of matchings of
G. This is equivalent to the number of matchings a graph contains, plus 1 to account
for the matching consisting of no edges.
Using this algorithm we determine the Hosoya index for pentane, iso-pentane and
neopentane. For pentane Hosoya index is 8. There is one way to choose zero edges,
4 ways to choose only one edge in the matching, and 3 ways to choose two edges in
the matching. This gives 1 + 4 + 3 = 8 simple matchings of pentane. Similarly for
iso-pentane is 1 + 4 + 2 = 7, and for neopentane 1 + 4 + 0 = 5.
5
CONCLUSION
The Wiener index is closely correlated with the boiling points of alkaline molecules.
The boiling point decreases in a general way as the compactness of the molecule increases, if the relative molecular mass is the same. The boiling point decreases with
increase branching. The quantitative structure-activity relationship showed that it is
also correlated with the critical point, the density, surface tension, and viscosity of its
liquid phase and vander Waals surface area of the molecule. Here we observed that,
the Wiener index, Hosoya index and Wiener Polynomial of isomers of C5 H12 are in the
order: pentane > iso-pentane > neopentane. Similarly the values of, density and boiling
point follow the same trend.
References
[1] A. T. Balaban, O. Ivanciuc, Amsterdam, (1999), 21-57.
[2] I. Gutman, S. Markovic, D. Lukovic, V. Radivojevic and S. Rancic, Coil. Sci.
Papers Fac. Sci. Kragujevac, 8,15 (1987), 2-15.
[3] H. Hosiya, Bull. Chem. Soc. Jpn. 4, (1971), 2332-2339.
[4] I. Lukvos, , J. Chem. Soc. Perkin, 2, 1667, (1988).
[5] S. Nikolic, N. Trinajstic and Z.Mihalic, Croat. Chem. Acta, 68, (1995), 105–129.
[6] D. H. Rouvray, J. Comput. Chem. 8, 470, (1987).
[7] D. H. Rouvray, W. Z. Tatong, Naturforsch, 41a, 1238, (1986).
[8] D. H. Rouvray, Chemical Graph Theory, Abacus Press, NewYork, (1991), 1-39.
[9] B. Wang, C. Ye, L. Van, Appl. Math. J. Chines Uni. 25, (2010), 155-161.
[10] B. Y. Yang, Y. N. Yeh, Proc. of Second International Taiwan-Moscow Algebra
Workshop, Tainan, (1997), 203–226.

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