On the extremal graphs with respect to the vertex PI index

The vertex Padmakar–Ivan (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 from u and v . We continue the research into estimating the extreme values of the PI index and answer the open question from [M.J. Nadjafi-Arani, G.H. Fath-Tabar, A.R. Ashrafi, Extremal graphs with respect to the vertex PI index, Appl. Math. Lett. 22 (2009) 1838–1840]. We prove that K ⌊ n / 2 ⌋ , ⌈ n / 2 ⌉ ′ , obtained from the complete bipartite graph K ⌊ n / 2 ⌋ , ⌈ n / 2 ⌉ by adding one edge connecting two vertices from the class of size ⌈ n / 2 ⌉ , is the unique graph with the second-maximal value of PI index. We also determine the structure of the extremal graphs that have second-minimal PI index n ( n − 1 ) + 2 among n -vertex graphs. In addition, we calculated PI index for all graphs on ⩽ 10 vertices.