关于树的核心中心与其他中心部分之间的距离
On the distances between the core center and other central parts of a tree
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中文总结 AI 辅助
研究树的核心中心与其他中心部分（如中心、质心、子树核心和特征中心）之间的距离，通过证明不等式给出距离上界，表明界是最优的，还得到使特征中心与核心中心距离最大的树并研究距离渐近行为。
中文摘要 AI 辅助
设\(T\)为一棵树。对于顶点\(v\in V(T)\)，离心子树数\(\epsilon_T(v)\)定义为\(\epsilon_T(v)=\min\{f_T(v,u): u\in V(T)\}\)，其中\(f_T(v,u)\)表示包含\(v\)和\(u\)的\(T\)的子树数量。\(T\)的核心顶点是具有最大离心子树数的顶点，\(T\)的所有核心顶点的集合称为\(T\)的核心中心。\(T\)的核心中心由单个顶点或两个相邻顶点组成。树中还有其他中心概念，如中心、质心、子树核心和特征中心，它们可能都不同。通过\(d_T(C, \mathfrak{C})\)、\(d_T(C_d, \mathfrak{C})\)和\(d_T(S_c, \mathfrak{C})\)分别表示\(T\)中中心与核心中心、质心与核心中心以及子树核心与核心中心之间的距离。我们证明对于\(n\geq 6\)个顶点的任何树\(T\)，有相应距离的不等式成立，且这些界是最优的，通过得到达到这些界的树来证明。还得到了在\(n\geq 6\)个顶点的所有树中使特征中心与核心中心之间距离最大的树，并研究了所有这些距离的渐近行为。
英文摘要
Let $T$ be a tree. For a vertex $v\in V(T)$, the eccentric subtree number $ε_T(v)$ is defined as $ε_T(v)=\min\{f_T(v,u): u\in V(T)\}$ where $f_T(v,u)$ denotes the number of subtrees of $T$ containing both $v$ and $u$. A core vertex of $T$ is a vertex with the maximum eccentric subtree number, and the set of all the core vertices of $T$ is called the core center of $T$. The core center of $T$ consists of either a single vertex or two adjacent vertices. There are other central concepts in a tree, such as the center, centroid, subtree core, and characteristic center, and these may all be different. By $d_T(C, \mathfrak{C})$, $d_T(C_d, \mathfrak{C})$ and $d_T(S_c, \mathfrak{C})$ we mean the distance between center and core center, distance between centroid and core center and distance between subtree core and core center in $T$, respectively. We show that for any tree $T$ on $n\geq 6$ vertices, (i)]$d_T(C,\mathfrak{C})\leq \lfloor \frac{n-g_0-4}{2} \rfloor$; (ii)]$d_T(C_d,\mathfrak{C})\leq \lfloor \frac{n-5}{2} \rfloor$; (iii)] $d_T(S_c,\mathfrak{C})\leq\left\{ \begin{array}{ll} 1, &\text{if $n=7$,} n-g_0-3, &\text{if $n\neq 7$;}\\ \end{array} \right.$ where $g_0\geq 2$ be the smallest positive integer such that $2^{g_0-1}+g_0\geq n-3$. Moreover, we show that these bounds are best possible by obtaining a tree which attains these bounds. We also obtain a tree which maximizes the distance between characteristic center and core center over all trees on $n\geq 6$ vertices. The asymptotic behaviour of all these distances are also studied.
