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.

Examples

The PI index of Dendrimer Nanostar of the following figure can be calculated by^[2]

${\displaystyle \operatorname {PI} (G_{n})=441\cdot 4^{n}-639\cdot 2^{n}+232,\quad n\geq 0.}$