Ashwini Index of a ‎Graph

Motivated by the terminal Wiener index‎, ‎we define the Ashwini index \mathcal{A} of trees as‎ \begin{eqnarray*}‎ % ‎\nonumber to remove numbering (before each equation)‎ ‎\mathcal{A}(T) &=& \sum\limits_{۱\leq i