Topological Index Calculator - Sample Examples

Table of Contents:

• Adjacency Matrix
• Distance Matrix
• Vertex Degree Vector
• Polarity Index
• Wiener Index
• Hosoya Index
• Randic Index
• Balaban Index
• Harary Index
• Vertex Degree Distance Index
• Odd-Even Index

Adjacency Matrix:

The adjacency matrix of a labeled connected graph G with N vertices is a square
symmetric matrix of order N. This matrix is defined below.



Example: 2,3-dimethylpentane



The adjacency matrix of G:

Distance Matrix:

The distance matrix of a labeled connected graph G with N vertices is a square
symmetric matrix of order N. This matrix is defined below.



where l_ij is the length of the shortest path (the distance) between the
vertices i and j in G.

Example: 2,3-dimethylpentane



The distance matrix of G:

Vertex Degree Vector:

The vertex degree vector of a labeled connected graph G with N vertices
is an (1 x N) vector. This vector is defined below.



Example: 2,3-dimethylpentane



The vertex degree of G:



The Vertex Degree Vector of G:

Polarity Index:

The Polarity Index was introduced by Wiener in 1947 as the half the sum of all
the distance matrix entries with a value of 3. This index is defined below.



where N is the number of vertices in the molecular graph.

Example: 2,3-dimethylpentane





The Polarity Index of G:

P(G) = (12 / 2) = 6

Wiener Index:

The Wiener Index was introduced by Wiener in 1947 as the path number.
This topological index is defined as the half-sum of the elements of the distance matrix.



where N is the number of vertices in the molecular graph.

Example: 2,3-dimethylpentane





The Wiener Index of G:

W(G) = (15 + 10 + 9 + 12 + 17 + 15 + 14) / 2 = 92 / 2 = 46

Hosoya Index:

The Hosoya Index was introduced by Hosoya in 1971 as the Z index.
This index is defined below.



where N is the number of vertices in the molecular graph and
p(G;i) is the number of selections of i mutually non-adjacent edges in G.
By definition, p(G;0) = 1, and p(G;1) is the number of edges in G.

Example: 2,3-dimethylpentane



p(G;0) = 1
p(G;1) = 6
p(G;2) = 8



p(G;3) = 2



The Hosoya Index of G:

Z(G) = p(G;0) + p(G;1) + p(G;2) + p(G;3) = 1 + 6 + 8 + 2 = 17

Randic Index:

The Randic Index was introduced by Randic in 1975 as the connectivity index.
This index is defined below.



where d(i) and d(j) are the valencies of the vertices i and j that define the edge ij

Example: 2,3-dimethylpentane



The count of the edge-types(the numbers at the vertices represent their valencies)



b₁₂ = 1
b₁₃ = 3
b₂₃ = 1
b₃₃ = 1

The Randic Index of G:

c(G) = 1*0.7071 + 3*0.5774 + 1*.4082 + 1*.3333 = 3.1807

Balaban Index:

The Balaban Index was introduced by Balaban in 1982 as the average-distance sum
connectivity. This index is defined below.



where M is the number of edges in G; m is the cyclomatic number of G;
and (D)_i is the distance sum where i = 1,2,...,N.

The cyclomatic number m = m(G) of a polycyclic graph G is equal to the
minimum number of edges that must be removed from G to transform it to
the related acyclic graph. For trees, m = 0; for monocycles, m = 1.

The distance sum (D)_i for a vertex i of G represents a sum of all entries
in the corresponding row of the distance matrix.

Example: 2,3-dimethylpentane





(D)₁ = 15
(D)₂ = 10
(D)₃ = 9
(D)₄ = 12
(D)₅ = 17
(D)₆ = 15
(D)₇ = 14

The Balaban Index of G:

J(G) = 6[2*(10*15)^-0.5 + (10*9)^-0.5 + (9*12)^-0.5 + (12*17)^-0.5 + (9*14)^-0.5] = 3.1442

Harary Index:

The Harary Index was introduced by Plavsic in 1991 in honor of Professor Frank Harary.
This index is defined below.



where N is the number of vertices in the molecular graph.

Example: 2,3-dimethylpentane







The Harary Index of G:

H(G) = .5(12*1 + 14/4 + 12/9 + 4/16) = 8.5417

Vertex Degree Distance Index:

The Vertex Degree Distance Index was introduced by Cao and Yuan in 2000 as
the degree of the vertices and the distance. This index is defined below.



where N is the number of vertices in the molecular graph and
f_i is the elements of vector (1xN) VD^-2 obtained by V multiply D^-2.

VD^-2 = [f₁,f₂,...,f_N]

Example: 2,3-dimethylpentane









The Vertex Degree Distance Index of G:

VDI(G) = (60739.101)^1/7 = 4.8234

Odd-Even Index: