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)=∑e=uv∈E(deg(u)+deg(v))(nu(e)+nv(e))PIw(G)=∑e=uv∈E(deg(u)+deg(v))(nu(e)+nv(e)), where deg(u)deg(u) denotes the vertex degree of uu and nu(e)nu(e) denotes the number of vertices of GG whose distance to the vertex uu is smaller than the distance to the vertex vv. We establish basic properties of PIw(G)PIw(G), and prove various lower and upper bounds. In particular, the path PnPn has minimal, while the complete tripartite graph Kn/3,n/3,n/3Kn/3,n/3,n/3 has maximal weighed vertex PIPI index among connected graphs with nn 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.