Extremal graphs with respect to the vertex PI index

The vertex PI index of a graph G is the sum over all edges u v ∈ 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 ) ≤ P I v ( 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.