On the eccentric connectivity index of trees with given domination number

Let $G$ be a simple connected finite graph. The eccentric connectivity index (ECI) of $G$ is defined as ${\xi }^c\left(G\right)={\sum }_{v\in V\left(G\right)}{\varepsilon }_G\left(v\right)d_G\left(v\right)$, where ${\varepsilon }_G\left(v\right)$ is the eccentricity of $v$, $d_G\left(v\right)$ is the degree of $v$. We denote the set of trees with order $n$ and domination number $\gamma$ by $T_{n,\gamma }$. In this paper, the extremal trees having the minimal ECI among $T_{n,\gamma }$ are determined. The tree among $T_{n,\gamma }$ satisfying $2\le \gamma \le \lceil \frac{n}{3}\rceil$ having the maximal ECI is also characterized. For $\lceil \frac{n}{3}\rceil \le \gamma \le \frac{n}{2}$, the tree among all caterpillars with domination number $\gamma$ having the maximal ECI is determined.