Eccentricity Sums in Trees

TL;DR

This paper investigates extremal properties of total eccentricity in trees, characterizing structures that maximize or minimize eccentricity ratios and total eccentricity given degree sequences.

Contribution

It provides new characterizations of extremal tree structures for eccentricity ratios and total eccentricity based on degree sequences.

Findings

Identified extremal trees for eccentricity ratios involving leaves and center.

Determined trees that minimize and maximize total eccentricity for given degree sequences.

Abstract

The eccentricity of a vertex, $ecc_T(v) = \max_{u\in T} d_T(v,u)$, was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and $ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of $T$. In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence.