Hosoya index

K4 matchings

Hosoya index

The Hosoya index, also known as the Z-index, is a topological index used in chemical graph theory and mathematical chemistry. It is named after the Japanese chemist Haruo Hosoya, who introduced it in 1971. The Hosoya index is a measure of the complexity of a molecular graph and is used to predict various chemical properties of molecules.

Definition

The Hosoya index of a graph \( G \) is defined as the total number of matchings in the graph, including the empty matching. In other words, it is the sum of the number of matchings of all sizes in the graph. Formally, if \( m_k(G) \) denotes the number of matchings of size \( k \) in \( G \), then the Hosoya index \( Z(G) \) is given by:

\[ Z(G) = \sum_{k=0}^{\lfloor n/2 \rfloor} m_k(G) \]

where \( n \) is the number of vertices in the graph \( G \).

Calculation

To calculate the Hosoya index, one must enumerate all possible matchings in the graph. This can be computationally intensive for large graphs, but efficient algorithms and software tools have been developed to facilitate this process.

Applications

The Hosoya index is used in various fields, including:

• Chemoinformatics: To predict the boiling point, melting point, and other physicochemical properties of organic compounds.
• Pharmacology: To study the structure-activity relationship (SAR) of drug molecules.
• Molecular biology: To analyze the structural properties of biomolecules such as proteins and nucleic acids.

Examples

For simple graphs, the Hosoya index can be calculated manually. For example:

• The Hosoya index of a path graph \( P_n \) with \( n \) vertices is the \( n \)-th Fibonacci number.
• The Hosoya index of a complete graph \( K_n \) is \( 1 \), as the only matching is the empty matching.

Related Concepts

• Topological index
• Wiener index
• Randić index
• Graph theory
• Chemical graph theory

See Also

• Haruo Hosoya
• Molecular graph
• Matching (graph theory)
• Chemoinformatics
• Structure-activity relationship