Center, centroid and subtree core of trees

TL;DR

This paper investigates the positions of the center, centroid, and subtree core in trees, identifying specific path-star trees that maximize the distances between these key vertices.

Contribution

It characterizes the trees that maximize the distances between the center, centroid, and subtree core, specifically identifying the extremal path-star trees.

Findings

Path-star trees maximize the distances between key vertices.

The tree P_{n-g_0,g_0} uniquely maximizes these distances.

The results apply to all trees with at least 5 vertices.

Abstract

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 degree $1$ vertex of the path $P_{n-g}$. We call the trees $P_{n-g,g}$ as path-star trees. We prove that over all trees on $n\geq 5$ vertices, the distance between center and subtree core and the distance between centroid and subtree core are 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 such that $2^{g_0}+g_0>n-1.$