^{}Department of Mathematics, Rani Channamma University, Belagavi, India.

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) &=& \sum\limits_{1\leq i<j\leq 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 $uv\in E(T)$. 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 $D_{h}$.