Extremal values on the eccentric distance sum of trees

Abstract: Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) = \sum_{u\in V_G}d_G(u,v)$ is the sum of all distances from the vertex $v$. 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.