On extremal leaf status and internal status

Abstract

For a vertex u of a tree T, the leaf (internal, respectively) status of u is the sum of the distances from u to all leaves (internal vertices, respectively) of T. The minimum (maximum, respectively) leaf status of a tree T is the minimum (maximum, respectively) leaf statuses of all vertices of T. The minimum (maximum, respectively) internal status of a tree T is the minimum (maximum, respectively) internal statuses of all vertices of T. We characterize those trees with the smallest (largest, respectively) extremal (minimum and maximum) leaf status and extremal (minimum and maximum) internal status, respectively. We also study the corresponding extremal problems for trees with given parameters, including diameter or maximum degree.