On the relation between Wiener index and eccentricity of a graph

The relation between the Wiener index W(G) and the eccentricity ε(G) of a graph G is studied. Lower and upper bounds on W(G) in terms of ε(G) are proved and extremal graphs characterized. A Nordhaus–Gaddum type result on W(G) involving ε(G) is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference W(T) - ε(T) is minimized on caterpillars. An exact formula for W(T) - ε(T) in terms of the radius of a tree T is obtained. A lower bound on the eccentricity of a tree in terms of its radius is also given. Two conjectures are proposed. The first asserts that the difference W(G) - ε(G) does not increase after contracting an edge of G. The second conjecture asserts that the difference between the Wiener index of a graph and its eccentricity is largest on paths.