Ashwini Index of a ‎Graph

Abstract

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<jleq n} d_{_{T}}(v_{i}‎, ‎v_{j}) [deg_{_{T}}(N(u_{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 $D_{h}$‎.