Bounds on the Weighted Vertex PI Index of Cacti Graphs

Abstract

The weighted vertex PI index of a graph $G$ is defined by
$$PI_w(G)=\sum_{e=uv\in E(G)}(d_G(u)+d_G(v))(n_u(e|G)+n_v(e|G))$$
where $d_G(u)$ denotes the vertex degree of $u$ and
$n_u(e|G)$ denotes the number of vertices in $G$ whose distance to the
vertex $u$ is smaller than the distance to the vertex $v$.
A graph is a cactus if it is connected and all its blocks are either edges or cycles.
In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti
with $n$ vertices and $s$ cycles,
and completely characterize the corresponding extremal graphs.