Eccentricity Sums in Trees

Abstract

The eccentricity of a vertex, $ec{c}_T(v)={\max }_{u\in T}{d}_T(v,u)$, 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 $Ecc(T)/ec{c}_T(u)$, $Ecc(T)/ec{c}_T(v)$, $ec{c}_T(u)/ec{c}_T(v)$, and $ec{c}_T(u)/ec{c}_T(w)$ 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.