Extremal values on the eccentric distance sum of trees

TL;DR

This paper investigates extremal values of the eccentric distance sum in trees, identifying those with minimal or maximal sums under various constraints such as domination number, number of leaves, and bipartition.

Contribution

It determines trees with extremal eccentric distance sums for different parameters, providing sharp bounds and characterizations for various classes of trees.

Findings

Identified trees with minimal eccentric distance sum for given domination number.

Determined trees with maximal eccentric distance sum when n= k*γ for specific k values.

Established sharp bounds on eccentric distance sums for trees with a fixed number of leaves.

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.