Descriptors for Organometallic Complexes - Fundamental QSARs for Metal Ions - page 115

on column j from the vertex on line i of the matrix. In fact, topological indices
are mathematical functions where the variables are values in the adjacency and
distances matrices.
As an example, we present the calculation of the topological indices W (Hosoya
1971), χ (Randic 1975), J (Balaban 1982), and MSD (Balaban 1983), chosen here
because they are very simple formulas/procedures of calculation.
The Wiener index (W) is calculated as the half-sum of all topological distances
collected in the distances matrix.

N N

1
2

∑ ∑

W =⋅ d ij (4.26)

i=1 j=1

The Randic index (χ) is the sum of the μ terms. Let δ be the degree of a vertex of
the graph and then define δ i and δ j as the degrees of two vertices that are connected.
The link (edge) between these vertices is characterized by μ (the inverse of the geo-
metrical mean of the degrees). The sum of all μ values is the index Randic χ.

μ = (δ i · δ j ) −1/2 (4.27)

χ = Σ μ i (4.28)

For the guanine molecule, χ = 5.2709 is calculated.
The Balaban index J is computed using the following procedure. Let N be the
number of vertices and L the number of edges of a certain kenograph. A square
matrix M that has N lines is built. Each element of the matrix is the topological dis-
tance between the vertex i and vertex j . Consequently, any line k and column k of the
matrix will be identical, and the diagonal will include null values.

Atom M ij Σ d j

1C 0 1 2 3 3 3 2 1 2 1 3 21

2N 1 0 1 2 4 4 3 2 3 2 2 24

3C 2 1 0 1 3 4 4 3 2 3 1 24

4N 3 2 1 0 2 3 3 2 1 4 2 23

5C 2 3 2 1 1 2 2 1 0 3 3 20

6C 1 2 3 2 2 2 1 0 1 2 4 20

7N 2 3 4 3 2 1 0 1 2 3 5 26

8C 3 4 4 3 1 0 1 2 2 4 5 29

9N 3 4 3 2 0 1 2 2 1 4 4 26

10O 1 2 3 4 4 4 3 2 3 0 4 30

11N 3 2 1 2 4 5 5 4 3 4 0 33

For the guanine molecule indexed as in Figure 4.5 (N = 11 and L = 12), the matrix
M is obtained. The vertex with index j has a value on the line j of the matrix. For each
vertex, the measure Σd j is calculated as a sum of the elements on the corresponding
line j of the matrix. For each link connecting two vertices i and j , μ is calculated, and
then the Balaban index (J) is calculated. For guanine, J = 2.0404 is calculated.