Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as \(\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)\). We show that the average eccentricity of a connected graph of order n, minimum degree \(\delta \) and maximum degree \(\Delta \) does not exceed \(\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7\), and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.

中文翻译：

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图形中的平均偏心度，最小度和最大度

令G为顶点集为V（G）的连通有限图。顶点v的离心率e（v）是从v到最远离v的顶点的距离。G的平均偏心率定义为\（\ frac {1} {| V（G）|} \ sum _ {v \ in V（G）} e（v）\）。我们表明，阶为n，最小度\（\ delta \）和最大度\（\ Delta \）的连通图的平均偏心率不超过\（\ frac {9} {4} \ frac {n- \ Delta -1} {\ delta +1} \ big（1 + \ frac {\ Delta-\ delta} {3n} \ big）+ 7 \），并且该边界与加法常数是尖锐的。我们为无三角形图和不包含4个循环的图提供了改进的边界。