Padmakar–Ivan index

Summary

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