Abstract The eccentricity of a vertex is the greatest distance from it to any other vertex and the average eccentricity of a graph G is the average value of eccentricities of all vertices of G . The proximity of a vertex in a connected graph is the average distance from it to all other vertices and the proximity of a connected graph G is the minimum average distance from a vertex of G to all others. A set S ⊆ V ( G ) is called a dominating set of G if N G ( x ) ⋂ S ≠ 0 for any vertex x ∈ V ( G ) ∖ S . The domination number γ ( G ) of G is the minimum cardinality of all dominating sets of G . In this paper, we improve and prove two AutoGraphiX conjectures. One gives the sharp upper bound on the quotient of the domination number and average eccentricity, and another shows the sharp upper bound about the difference between the domination number and proximity.

中文翻译：

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AutoGraphiX 关于支配数、平均偏心率和接近度猜想的证明

摘要 一个顶点的偏心率是它到任何其他顶点的最大距离，图G的平均偏心率是G的所有顶点偏心率的平均值。连通图中顶点的接近度是它到所有其他顶点的平均距离，连通图 G 的接近度是 G 的一个顶点到所有其他顶点的最小平均距离。如果对于任何顶点 x ∈ V ( G ) ∖ S NG ( x ) ⋂ S ≠ 0，则集合 S ⊆ V ( G ) 称为 G 的支配集。G 的支配数 γ ( G ) 是 G 的所有支配集的最小基数。在本文中，我们改进并证明了两个 AutoGraphiX 猜想。一个给出了支配数和平均偏心率的商的尖锐上限，