Eccentricity Sums in Trees

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.