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The vertex PI index of a graph $G$ is the sum over all edges $uv\in E(G)$ of the number of vertices which are not equidistant to $u$ and $v$. In this paper, the extremal values of this new topological index are computed. In particular, we prove that for each $n$-vertex graph $G,n(n-1)\leq PI_{v}(G)\leq n.[\frac{n}{2}].[\frac{n}{2}]$ , where $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$ and $\lceil x\rceil$ is the smallest integer not less than $x$. The extremal graphs with respect to the vertex PI index are also determined.
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