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:
A ij
= 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:
D ij
= l ij
; if i ≠ j
= 0; otherwise where l ij 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∑ ∑ D ij i=1 i=1
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.
ethane propane
2-methylpropane butane
2,2-dimethylpropane
2-methylbutane pentane
2,2-dimethylbutane
.
.
2,3-dimethylbutane
2-methylpentane
3-methylpentane hexane
4-ethyl-2-methylhexane
3,5-dimethylheptane
2,5-dimethylheptane
2,6-dimethylheptane
2-methyloctane
N: Polarity: Wiener:
2
3
4
4
5
6
0.0000
0.0000
0.0000
1.0000
4.0000
9.0000
1.0000 10.0000
0.0000 16.0000
5 2.0000 18.0000
5 2.0000 20.0000
3.0000 28.0000
6 4.0000 29.0000
6 3.0000 32.0000
6 4.0000 31.0000
6 3.0000 35.0000
9
9
9
9
9
.
.
.
.
8.0000 98.0000
8.0000 100.0000
7.0000 104.0000
6.0000 108.0000
6.0000 114.0000
3-methyloctane
4-methyloctane
3-ethylheptane
4-ethylheptane
9
9
9
9
7.0000 110.0000
7.0000 108.0000
8.0000 104.0000
8.0000 102.0000
nonane 9 6.0000 120.0000
2,2,3,3,4-pentamenthylpentane 10 15.0000 108.0000
.
.
Balaban: Odd-even:
1.0000
2.0000
VDI:
1.0000
Harary:
1.0000
Randic:
1.0000
1.6330
2.3238
3.5000
2.1634
2.2500
1.4142
4.5000
3.3677
3.7500
1.7321
1.9747
3.0237
2.5395
2.1906
3.1685
5.2222
5.0000
2.9131
4.5895
3.6111
5.5000
1.9142
2.0000
6.4444
3.8514
5.2222
2.2701
6.8194
3.4010
5.0347
2.4142
7.1667
4.8758
7.0833
2.5607
2.9935
2.6272
2.7542
2.3391
.
.
7.8889
7.9167
8.2639
8.4967
.
.
4.6112
4.1451
4.1925
3.7591
.
.
6.9444
6.7083
6.7569
6.4983
.
.
2.6427
2.7701
2.8081
2.9142
3.3074
12.7228
5.0925 11.7364
4.2019
3.2230
12.7922
5.0849 11.7017
4.2019
3.0608
12.7194
5.0562 11.5958
4.1639
2.9147
12.3061
5.0290 11.5003
4.1259
2.7467
12.9433
4.7280 11.2633
4.2701
.
.
2.8766
13.1247
4.7549 11.3540
4.3081
2.9548
13.0553
4.7479 11.3887
4.3081
3.0922
13.2367
4.7903 11.4794
4.3461
3.1753
13.1672
4.7880 11.5142
4.3461
2.5951
13.4120
4.4434 11.0289
4.4142
4.4038
13.0833
6.6909 14.7917
4.1934
ethane propane
2-methylpropane butane
2,2-dimethylpropane
2-methylbutane pentane
2,2-dimethylbutane
N: Polarity: Wiener:
2
3
4
0.0000
0.0000
0.0000
1.0000
4.0000
9.0000
4 1.0000 10.0000
5 0.0000 16.0000
5 2.0000 18.0000
5 2.0000 20.0000
6 3.0000 28.0000
2,3-dimethylbutane
2-methylpentane
3-methylpentane
.
hexane
.
4-ethyl-2-methylhexane
6 4.0000 29.0000
6 3.0000 32.0000
6 4.0000 31.0000
6 3.0000 35.0000
.
.
.
.
.
9 8.0000 98.0000
.
3,5-dimethylheptane
2,5-dimethylheptane
2,6-dimethylheptane
2-methyloctane
9 8.0000 100.0000
9 7.0000 104.0000
9 6.0000 108.0000
9 6.0000 114.0000
3-methyloctane
4-methyloctane
3-ethylheptane
4-ethylheptane
9
9
9
9
7.0000 110.0000
7.0000 108.0000
8.0000 104.0000
8.0000 102.0000
nonane 9 6.0000 120.0000
2,2,3,3,4-pentamenthylpentane 10 15.0000 108.0000
Balaban: Odd-even:
1.0000
1.6330
2.3238
2.0000
3.5000
4.5000
VDI:
1.0000
2.1634
3.3677
Harary:
1.0000
2.2500
3.7500
Randic:
1.0000
1.4142
1.7321
1.9747
3.0237
2.5395
2.1906
3.1685
5.2222
5.0000
2.9131
4.5895
3.6111
5.5000
1.9142
2.0000
6.4444
3.8514
5.2222
2.2701
6.8194
3.4010
5.0347
2.4142
7.1667
4.8758
7.0833
2.5607
2.9935
2.6272
2.7542
7.8889
4.6112
6.9444
2.6427
7.9167
4.1451
6.7083
2.7701
8.2639
4.1925
6.7569
2.8081
2.3391
.
8.4967
3.7591
6.4983
2.9142
.
.
.
.
.
.
.
.
3.3074
12.7228
5.0925 11.7364
4.2019
.
3.2230
12.7922
5.0849 11.7017
4.2019
3.0608
12.7194
5.0562 11.5958
4.1639
2.9147
12.3061
5.0290 11.5003
4.1259
2.7467
12.9433
4.7280 11.2633
4.2701
2.8766
13.1247
4.7549 11.3540
4.3081
2.9548
13.0553
4.7479 11.3887
4.3081
3.0922
13.2367
4.7903 11.4794
4.3461
3.1753
13.1672
4.7880 11.5142
4.3461
2.5951
13.4120
4.4434 11.0289
4.4142
4.4038
13.0833
6.6909 14.7917
4.1934
Adjacency Matrix: 2,3-dimethylbutane
A
0 1 0 0 0 0
1 0 1 0 1 0
= 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
D
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
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
Index equations created for particular indices to predict approximate boiling point of molecules:
N:
Polarity:
Wiener:
Balaban:
Odd-Even:
Vertex Degree Distance:
Harary:
Randic:
BP = 177.38 ln(N) + 24.742
BP = 10.16 (Polarity Index) + 323.6
BP = 25.684 (Balaban Index) + 324.95
BP = 45.453 ln(VDD Index) + 313.74
BP = 15.036 (Harary Index) + 220.98
BP = 184.73 ln(Randic Index) + 150.09
Average Error: 2.30%
Average Error: 4.63%
BP = 56.81 ln(Wiener Index) + 157.99
Average Error: 2.85%
Average Error: 7.69%
BP = 156.97 ln(Odd-Even Index) + 1.9792
Average Error: 4.53%
Average Error: 8.95%
Average Error: 3.21%
Average Error: 1.45%
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
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.