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Topological Index Calculator
A JavaScript application to introduce
quantitative structure-property relationships
(QSPR) in undergraduate organic chemistry Irvin
J. Levy, Departments of Chemistry Computer
Science, Gordon College, Wenham, MA 01984,
ijl_at_gordon.edu Steven D. Granz, Departments of
Mathematics Computer Science, Gordon College
Results
Abstract
Background
Since the development of the Wiener Index,
numerous topological indices have been described.
These methods convert molecular structure to
a mathematical representation (a chemical graph)
and then define computations to be performed on
the resulting graph. Statistical
correlations between those results and physical
properties serve as a predictive tool. In
organic chemistry, students are taught the
relationship between molecular structure and
boiling point but generally do not investigate
the phenomenon because tools to support the
tedious calculations are lacking. We have
developed a JavaScript program, "Topological
Index Calculator," which computes key indices
rapidly. Use of JavaScript benefits instructors
who may wish to modify or extend the program's
capabilities and students who may want to use the
tool easily both in and out of the laboratory.
With this program, students may work
cooperatively to develop correlations between
topological indices and physical properties of
alkanes.
A topological index is a value that is dependent
on the molecular structure of a molecule. They
are used to approximate physical properties of
molecules, such as the boiling point. To get a
better understanding of how indices are used, we
will examine how to calculate the Wiener Index of
a molecule. Two very important
graph-theoretical matrices are the adjacency
matrix and the distance matrix. Both of these
can be used to find the Weiner Index of a
molecule. The adjacency matrix A of a labelled
connected graph G with N vertices, is a square
symmetric matrix of order N. It is defined
as Aij 1 if vertices i and j are
adjacent 0 otherwise The
distance matrix D of a labelled connected graph
with N vertices, is a square symmetric matrix of
order N. It is defined as Dij lij if i ?
j 0 otherwise where lij is the length of
the shortest path (the distance) between the
vertices i and j in G. The Wiener Index is
defined as one-half the sum of the elements of
the distance matrix. N N W 1/2?
? Dij i1 i1 For example What is the
Wiener Index of 2,3-dimethylbutane?
Topological indices can be calculated quickly
using the Topological Index Calculator. This
information can easily be used to create an index
equation by plotting the experimental boiling
point vs. the index computed for a set of
molecules and performing a linear regression
analysis on the data. For example, data in the
table below can be used to generate index
equations for alkanes.
Adjacency Matrix 2,3-dimethylbutane 0 1
0 0 0 0 1 0 1 0 1 0 A
0 1 0 1 0 1 0 0 1 0 0
0 0 1 0 0 0 0 0 0 1
0 0 0 Distance Matrix 2,3-dimethylbutane
0 1 2 3 2 3 1 0 1 2 1
2 D 2 1 0 1 2 1 3 2 1
0 3 2 2 1 2 3 0 3 3
2 1 2 3 0 Wiener Index
2,3-dimethylbutane 0 1 2 3 2 3
1 0 1 2 1 2 2 1 0 1 2
1 3 2 1 0 3 2 2 1 2 3
0 3 3 2 1 2 3 0 58 Wiener
Index 58 / 2 29

• Future Directions
• use the tool to verify values found in the
  literaure
• develop new indices with better approximations
  of the boiling point
• combine current indices with one another
• develop unique index

Index equations created for particular indices to
predict approximate boiling point of molecules
N BP 177.38 ln(N) 24.742
Average Error 2.30 Polarity BP 10.16
(Polarity Index) 323.6 Average Error
4.63 Wiener BP 56.81 ln(Wiener Index)
157.99 Average Error 2.85 Balaban BP
25.684 (Balaban Index) 324.95 Average Error
7.69 Odd-Even BP 156.97 ln(Odd-Even
Index) 1.9792 Average Error 4.53 Vertex
Degree Distance BP 45.453 ln(VDD Index)
313.74 Average Error 8.95 Harary BP
15.036 (Harary Index) 220.98 Average
Error 3.21 Randic BP 184.73 ln(Randic
Index) 150.09 Average Error
1.45
References Cao, C. "Topological Indices
Based on Vertex, Distance and Ring On Boiling
Points of Paraffins and Cycloalkanes." J. Chem.
Inf. and Comp. Sci., 2001, 41, 4. Mihalic, Z.
"A Graph-Theoretical Approach to
Structure-Property Relationships." J. Chem. Educ.
1992, 69, 9. Trinajstic, N. Chemical Graph
Theory. Vol II. Florida CRC Press, 1983.

• http//www.math-cs.gordon.edu/courses/topo/