Eccentricity sums in trees

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