# Balaban J index

The Balaban-J-Index is a topological index named after Alexandru Balaban , which is characterized by being able to differentiate between similar molecular structures. This is referred to as a low tendency to degenerate . The J-index is determined from the distance matrix of the molecular structure, hydrogen not being taken into account.

## equation

${\ displaystyle J \; = \; {\ frac {q} {\ mu +1}} \ sum _ {i, j} ^ {n} {\ frac {1} {\ sqrt {s_ {i} s_ { j}}}}}$ With

q : number of bonds in the molecule

μ : number of rings in the molecule

${\ displaystyle \ mu \; = \; q-n + 1}$ With
n : number of atoms in the molecule

s i and s j : Sum of the weighted distances between the atom i or the atom j and all other atoms of the molecule. The distance is the number of bonds between the atoms. If there are several possible routes (through rings), the shortest route must be chosen. Single bonds are included by the factor 1, double bonds by a factor of 1 / 2 , triple bonds by a factor of 1 / 3 , and aromatic bonds by a factor of 2 / 3 .

Only connected (neighboring) atoms are taken into account for the sum . ${\ displaystyle \ textstyle \ sum _ {i, j} ^ {n} {\ frac {1} {\ sqrt {s_ {i} s_ {j}}}}}$ ## Sample calculation

The distance matrix is ​​a table that shows the number of bonds between two atoms in the molecule:

atom  1   2   3   4th   5   6th   7th   8th  Distance sum
s i
1 0 1 2 3 4th 5 3 4th 22nd
2 1 0 1 2 3 4th 2 3 16
3 2 1 0 1 2 3 1 2 12
4th 3 2 1 0 1 2 2 3 14th
5 4th 3 2 1 0 1 3 4th 18th
6th 5 4th 3 2 1 0 4th 5 24
7th 3 2 1 2 3 4th 0 1 16
8th 4th 3 2 3 4th 5 1 0 22nd
• The number of ties is q = 7
• Number of atoms is n = 8
• Number of rings is μ = 0 (= 7-8 + 1)

The sum results from ${\ displaystyle 1 / {\ sqrt {22 \ cdot 16}} \; + \; 1 / {\ sqrt {16 \ cdot 12}} \; + \; 1 / {\ sqrt {12 \ cdot 14}} \ ; + \; 1 / {\ sqrt {14 \ cdot 18}} \; + \; 1 / {\ sqrt {18 \ cdot 24}} \; + \; 1 / {\ sqrt {24 \ cdot 16}} \; + \; 1 / {\ sqrt {16 \ cdot 22}} = 0 {,} 4392}$ This results in the Balaban J index for ethylhexane as (not rounded) ${\ displaystyle J \; = \; 7 \ cdot 0 {,} 4392 = 3 {,} 07437}$ ## meaning

The J-index is characterized by the fact that it allows a very detailed distinction between similar structures. Due to this high ability to differentiate, the J-Index can be used successfully in quantitative structure-activity relationships (QSPR).

## literature

1. Alexandru T. Balaban : Highly Discriminating Distance-based Topological Index . In: Chemical Physics Letters . tape 89 , no. 5 , July 1982, p. 399-404 , doi : 10.1016 / 0009-2614 (82) 80009-2 ( elsevier.com ).