Discrete Applied Mathematics
The eccentricity of a vertex, eccT(v)=maxu∈TdT(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)/eccT(u), Ecc(T)/eccT(v), eccT(u)/eccT(v), and eccT(u)/eccT(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.
Smith, Heather, Laszlo A. Szekely, Hua Wang.
"Eccentricity Sum in Trees."
Discrete Applied Mathematics, 207: 120-131.
doi: 10.1016/j.dam.2016.02.013 source: https://www.sciencedirect.com/science/article/pii/S0166218X16300646?via%3Dihub