Maximizing distance between center, centroid and subtree core of trees

[摘要] For $n \geq 5$ and $2 \leq g \leq n−3$, consider the tree $P_{n−g,g}$ on $n$ vertices which is obtained by adding $g$ pendant vertices to one end vertex of the path $P_{n−g}$. We call the trees $P_{n−g,g}$ as path-star trees. The subtree core of a tree $T$ is the set of all vertices $v$ of $T$ for which the number of subtrees of $T$ containing $v$ is maximum. We prove that over all trees on $n \geq 5$ vertices, the distance between the center (respectively, centroid) and the subtree core is maximized by some path-star trees. We also prove that the tree $P_{n−g0,g0}$ maximizes both the distances among all path-star trees on $n$ vertices, where $g0$ is the smallest positive integer satisfying $2^{g0} + g0$ > $n − 1$.

[学科分类] 数学（综合）

[关键词] Tree;center;centroid;subtree core;distance