Extremal values on the eccentric distance sum of trees

Let $G=(V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as ${\xi }^d(G)={\sum }_{v\in V_G}{\epsilon }_G(v)D_G(v)$, where ${\epsilon }_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) = \sum_{u\in V_G}d_G(u,v)$isthesumofalldistancesfromthevertex$v$. In this paper the tree among $n$-vertex trees with domination number $\gamma$ having the minimal eccentric distance sum is determined and the tree among $n$-vertex trees with domination number $\gamma$ satisfying $n=k\gamma$ having the maximal eccentric distance sum is identified, respectively, for $k=2,3,\frac{n}{3},\frac{n}{2}$. Sharp upper and lower bounds on the eccentric distance sums among the $n$-vertex trees with $k$ leaves are determined. Finally, the trees among the $n$-vertex trees with a given bipartition having the minimal, second minimal and third minimal eccentric distance sums are determined, respectively.