Randic

Randic index, also written as the Randić index, is a topological descriptor used in chemical graph theory to quantify molecular branching and connectivity. It is named after Croatian chemist Milan Randić, who introduced the concept in the 1970s.

In graph terms, a molecule is modeled as a simple graph G = (V,E) where vertices represent atoms and edges represent bonds. Let d(u) denote the degree of vertex u. The classical Randic index R(G) is defined as the sum over all edges uv in E of 1 divided by the square root of the product of the degrees of the incident vertices: R(G) = sum_{uv in E} 1 / sqrt(d(u) d(v)).

Generalizations exist in the form of R_α(G) = sum_{uv in E} [d(u) d(v)]^α for real numbers α, with the classical Randic index corresponding to α = -1/2. Other common choices of α yield different sensitivity to branching and degree distribution.

Properties and uses: The Randic index reflects molecular branching, typically increasing with greater branching for a fixed number of atoms. It is additive over disjoint unions: R(G1 ∪ G2) = R(G1) + R(G2). The index has widespread application as a descriptor in QSAR and QSPR studies, contributing to correlations with properties such as boiling points, stability, and biological activity. It remains a standard tool in both theoretical chemistry and graph theory for analyzing molecular structure and connectivity.