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 weighted version defined as $PI_w (G) = \sum_{e = uv \in E} (deg (u) + deg (v)) (n_u (e) + n_v (e))$, 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 $PI_w (G)$, 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 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. 
Aleksandar Ili\' c and Nikola Milosavaljevi\' c
math.CO