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| Paper IPM / M / 11340 |
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| Abstract: | |
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The vertex PI index of a graph G is the sum over all edges uv ∈ 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) ≤ PIv(G) ≤ n.[[(n)/2]].[[(n)/2]] , where ⎣x⎦ denotes the greatest integer not exceeding x and ⎡x⎤ 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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