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On the Relation Between Wiener Index and Eccentricity of a Graph
The relation between the Wiener index and the eccentricity
of a graph is studied. Lower and upper bounds on in
terms of are proved and extremal graphs characterized. A
Nordhaus-Gaddum type result on involving 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 is minimized on caterpillars. An exact
formula for in terms of the radius of a tree 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
does not increase after contracting an edge of . The
second conjecture asserts that the difference between the Wiener index of a
graph and its eccentricity is largest on paths