title: Padmakar–Ivan index
text: # 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 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 $n_{eu}(e\mid G)$ denotes the number of edges lying closer to the vertex u than the vertex v, and $n_{ev}(e\mid G)$ 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
  
      $\operatorname {PI} (G)=\sum _{e\in E(G)}[n_{eu}(e\mid G)+n_{ev}(e\mid G)]$
  
  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
  
      $\operatorname {PI} (G_{n})=441\cdot 4^{n}-639\cdot 2^{n}+232,\quad n\geq 0.$
  