The weighted vertex PI index
The vertex PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. In order to increase diversity of this topological index for bipartite graphs, we introduce a weighted version defined as PIw(G) = Sigma(e=uv is an element of E) (deg(u) + deg(v))(n(u)(e) + n(v)(e)), where deg(u) denotes the vertex degree of u and n(u)(e) denotes the number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v. We establish basic properties of PIw(G), and prove various lower and upper bounds. In particular, the path P-n has minimal, while the complete tripartite graph K-n/3, (n/3), (n/3) has maximal weighed vertex PI index among connected graphs with n vertices. We also compute exact expressions for the weighted vertex PI index of the Cartesian product of graphs. Finally we present modifications of two inequalities and open new perspectives for the future research. (C) 2012 Elsevier Ltd. All rights reserved.
pi index cartesian product szeged index distance in graphs number of triangles international olympiad in mathematics