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) &=& sumlimits_{1leq i‎&+& deg_{_{t}}(n(v_{j}))],‎ ‎end{eqnarray*}‎ ‎where $d_{t}(v_{i}‎, ‎v_{j})$ is the distance between the vertices $v_{i}‎, ‎v_{j} in v(t)$‎, ‎is equal to the length of the shortest path starting at $v_{i}$ and ending at $v_{j}$ and $deg_{t}(n(v))$ is the cardinality of $deg_{t}(u)$‎, ‎where $uvin e(t)$‎. ‎in this paper‎, ‎trees with minimum and maximum $mathcal{a}$ are characterized and the expressions for the‎ ‎ashwini index are obtained for detour saturated trees $t_{3}(n)$‎, $t_{4}(n)$ as well as a class of dendrimers $d_{h}$‎.