Maximizing distance between center, centroid and subtree core of trees

Abstract

For $$n\ge 5$$ and $$2\le g\le n-3,$$ 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 $$n\ge 5$$ 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 $$P_{n-g_0,g_0}$$ maximizes both the distances among all path-star trees on n vertices, where $$g_0$$ is the smallest positive integer satisfying $$2^{g_0}+g_0>n-1$$ .