The relation between the Wiener index W(G) and the eccentricity \(\varepsilon (G)\) of a graph G is studied. Lower and upper bounds on W(G) in terms of \(\varepsilon (G)\) are proved and extremal graphs characterized. A Nordhaus–Gaddum type result on W(G) involving \(\varepsilon (G)\) 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 \(W(T) - \varepsilon (T)\) is minimized on caterpillars. An exact formula for \(W(T) - \varepsilon (T)\) 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 \(W(G) - \varepsilon (G)\) 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.

研究了图G的维纳指数W（G）与偏心率\（\ varepsilon（G）\）之间的关系。证明了W（G）的上下界（\（\ varepsilon（G）\））并刻画了极值图。给出了涉及\（\ varepsilon（G）\）的W（G）的Nordhaus–Gaddum类型结果。证明了从偏心率角度看树的Wiener指数有一个尖锐的上限。结果表明，在相同阶数的树中，毛虫的差异\（W（T）-\ varepsilon（T）\）最小。的精确公式获得就树T的半径而言的\（W（T）-\ varepsilon（T）\）。还给出了树的偏心度的下限（以其半径为单位）。提出了两个猜想。第一个断言，在缩小G的边缘后，差\（W（G）-\ varepsilon（G）\）不会增加。第二个猜想断言，图的维纳指数与其偏心率之间的差异在路径上最大。