**A topological index** is an invariant of a molecular graph in computer chemistry problems, a certain (usually numerical) value (or set of values) characterizing the structure of a molecule . Typically, topological indices do not reflect the multiplicity of chemical bonds and types of atoms (C, N, O, etc.), hydrogen atoms are not taken into account. The most famous topological indices include the Hosoi index, Wiener index, Randich index , Balaban index and others. ^{[1]} ^{[2]}

## Global and local indexes

The Hosoi Index and the Wiener Index are examples of global (or integral) topological indices that reflect the structure of a given molecule. Bonchev and Polyansky proposed a local (differential) index for each atom in the molecule. ^{[1]} As another example of local indices, modifications of the Hosoi index can be given. ^{[3]}

## Discrimination ability and super indexes

Values of the same topological index for several different molecular graphs may coincide. The fewer such matches, the higher the so-called discriminatory ability of the index. This ability is a critical index feature. To increase it, several topological indices can be combined into one super-index. ^{[four]}

## Computational complexity

Computational complexity is another important characteristic of a topological index. Many indices, such as the Wiener index, the Randic index and the Balaban index, are calculated using fast algorithms, unlike, for example, the Hosoi index and its modifications, for which only time-exponential algorithms are known. ^{[3]}

## Application

Topological indices are used in computer chemistry to solve a wide range of general and special problems. These tasks include: searching for substances with predetermined properties (searching for dependencies such as “ structure-property ”, “structure-pharmacological activity”), primary filtering of structural information for the repeated generation of molecular graphs of a given type, preliminary comparison of molecular graphs when they are tested for isomorphism and a number of others. The topological index depends only on the structure of the molecule, and not on its composition, therefore, molecules of the same structure (at the level of structural formulas ), but of different composition, for example, furan and thiophene will have equal indices. To overcome this difficulty, a number of indices were proposed, for example, electronegativity indices. ^{[five]}

## See also

- Graph invariant

## Notes

- ↑
^{1}^{2}Chemical Applications of Topology and Graph Theory, ed. R. King = Chemical Applications of Topology and Graph Theory, ed. by RB King. - M .: Mir, 1987 .-- 560 p. - ↑
*M.I. Stankevich, I.V. Stankevich, N.S. Zefirov .*Topological indices in organic chemistry (rus.) // Advances in Chemistry . - Russian Academy of Sciences , 1988. - V. 57 , No. 3 . - S. 337-366 . - ↑
^{1}^{2}Trofimov MI, An Optimization of Procedure for Calculation of Hosoya's Index, J. Math. Chem., 1991, 8, 327. - ↑ Bonchev D., Mekenyan O., Trinajstic N., J. Comp. Chem., 1981, 2, 127
- ↑ M. I. Trofimov, E. A. Smolensky,
*Bulletin of the Academy of Sciences.**Chemical Series*, 2005, 2166-2176.