发表机构
P.J. Šafárik University in Košice（科希策帕尔詹·沙法里克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图的介数中心与外围，给出其位于单一图块的充分条件，证明任意图可作为某图的介数中心子图，并证明树的介数中心含于路径，且枚举了至多20阶的树。
AI 中文摘要
图 $G = (V,E)$ 中顶点 $v$ 的介数中心性是通过 $v$ 的 $G$ 的最短路径的相对数量之和。$G$ 中具有最大（相应地，最小）介数中心性的顶点诱导出 $G$ 的介数中心（相应地，介数外围）。我们研究图的介数中心性及其在图块中的定位，给出了图（根据直径或块大小）具有这些中心包含在单个块中的充分条件。进一步，我们证明每个图都作为某个图的介数中心所诱导的子图出现（也作为介数外围所诱导的子图出现）。对于树，我们通过另一种证明方法，证明其介数中心总是包含在一条路径中；此外，我们根据介数中心的阶数枚举了阶数至多为20的树。
英文摘要
The betweenness centrality of a vertex $v$ in a graph $G = (V,E)$ is the sum of the relative numbers of shortest paths of $G$ that pass through $v$. The vertices of $G$ which have the maximum (resp. minimum) betweenness induce the betweenness center (resp. betweenness periphery) of $G$. We study betweenness of graphs and their localization in graph blocks, presenting sufficient conditions for graphs (in terms of diameter or block sizes) to have those centers contained in a single block. Further, we show that each graph occurs as the subgraph induced by the betweenness center of some graph (as well as the subgraph induced by the betweenness periphery). For trees, we show, by an alternative proof, that their betweenness center is always contained in a path; in addition, we enumerate trees of order at most 20 according to the order of their betweenness centers.
