Abstract We introduce the general eccentric connectivity index of a graph G , E C I α ( G ) = ∑ v ∈ V ( G ) e c c G ( v ) d G α ( v ) for α ∈ R , where V ( G ) is the vertex set of G , e c c G ( v ) is the eccentricity of a vertex v and d G ( v ) is the degree of v in G . We present lower and upper bounds on the general eccentric connectivity index for trees of given order, trees of given order and diameter, and trees of given order and number of pendant vertices. Then we give lower and upper bounds on the general eccentric connectivity index for unicyclic graphs of given order, and unicyclic graphs of given order and girth. The upper bounds for trees of given order and diameter, and trees of given order and number of pendant vertices hold for α > 1 . All the other bounds are valid for 0 α ≤ 1 or 0 α 1 . We present all the extremal graphs, which means that our bounds are best possible.

中文翻译：

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树和单环图的一般偏心连通性指数

摘要 我们介绍了图 G 的一般偏心连通性指数，ECI α ( G ) = ∑ v ∈ V ( G ) ecc G ( v ) d G α ( v ) for α ∈ R ，其中 V ( G ) 是顶点G 的集合，ecc G(v) 是顶点 v 的偏心率，d G(v) 是 v 在 G 中的度数。我们给出了给定顺序的树、给定顺序和直径的树以及给定顺序和悬垂顶点数量的树的一般偏心连通性指数的下限和上限。然后我们给出给定阶次的单环图和给定阶次和周长的单环图的一般偏心连通性指数的下限和上限。给定顺序和直径的树以及给定顺序和悬垂顶点数量的树的上限适用于 α > 1 。所有其他边界对于 0 α ≤ 1 或 0 α 1 都有效。我们展示了所有的极值图，