Padmakar–Ivan index

In chemical graph theory, the Padmakar–Ivan (PI) index is a topological index of a molecule, used in biochemistry. The Padmakar–Ivan index is a generalization introduced by Padmakar V. Khadikar and Iván Gutman[1] of the concept of the Wiener index, introduced by Harry Wiener. The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. Let G be a graph and e = uv an edge of G. Here ${\displaystyle n_{eu}(e\mid G)}$ denotes the number of edges lying closer to the vertex u than the vertex v, and ${\displaystyle n_{ev}(e\mid G)}$ is the number of edges lying closer to the vertex v than the vertex u. The Padmakar–Ivan index of a graph G is defined as

${\displaystyle \operatorname {PI} (G)=\sum _{e\in E(G)}[n_{eu}(e\mid G)+n_{ev}(e\mid G)]}$

The PI index is very important in the study of quantitative structure–activity relationship for the classification models used in the chemical, biological sciences, engineering, and nanotechnology.