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Extremal trees of a given degree sequence or segment sequence with respect to average Steiner 3-eccentricity
Applied Mathematics and Computation ( IF 3.5 ) Pub Date : 2022-09-28 , DOI: 10.1016/j.amc.2022.127556
Shuchao Li , Xin Liu , Wanting Sun , Lixia Yan

The Steiner k-eccentricity of a vertex in a graph G is the maximum Steiner distance over all k-subsets containing the vertex. The average Steiner k-eccentricity of G is the mean value of all vertices’ Steiner k-eccentricities in G. Let Tn be the set of all n-vertex trees, Tn,Δ be the set of n-vertex trees with maximum degree Δ, Tn,Δk be the set of n-vertex trees with exactly k vertices of a given maximum degree Δ, and let MTnk be the set of n-vertex trees with exactly k vertices of maximum degree. In this paper, we first determine the sharp upper bound on the average Steiner 3-eccentricity of n-vertex trees with a given degree sequence. The corresponding extremal graphs are characterized. Consequently, together with majorization theory, all graphs among Tn,Δk (resp. Tn,Δ,MTnk,Tn) having the maximum average Steiner 3-eccentricity are identified. Then we characterize the unique n-vertex tree with a given segment sequence having the minimum average Steiner 3-eccentricity. Finally, we determine all n-vertex trees with a given number of segments having the minimum average Steiner 3-eccentricity.



中文翻译:

给定度数序列或段序列相对于平均施泰纳 3 偏心率的极值树

施泰纳ķ- 图中顶点的偏心率G是最大的施泰纳距离ķ- 包含顶点的子集。平均斯坦纳ķ- 偏心率G是所有顶点的 Steiner 的平均值ķ- 偏心度G. 让n成为所有的集合n- 顶点树,n,Δ是一组n- 具有最大度数的顶点树Δ,n,Δķ是一组n- 完全具有的顶点树ķ给定最大度数的顶点Δ, 然后让公吨nķ是一组n- 完全具有的顶点树ķ最大度数的顶点。在本文中,我们首先确定了平均 Steiner 3 偏心率的急剧上限n-具有给定度数序列的顶点树。表征相应的极值图。因此,连同大数化理论,所有图n,Δķ(分别。n,Δ,公吨nķ,n) 具有最大平均 Steiner 3 偏心距。然后我们表征独特的n-具有给定分段序列的顶点树,具有最小平均 Steiner 3 偏心率。最后,我们确定所有n- 具有给定段数的顶点树,具有最小平均 Steiner 3 偏心率。

更新日期:2022-09-28
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