On extremal leaf status and internal status of trees

TL;DR

This paper investigates extremal values of leaf and internal vertex statuses in trees, characterizing cases with given diameter or maximum degree, and provides bounds for these parameters.

Contribution

It determines the smallest and largest possible values of leaf and internal statuses in trees and characterizes the extremal structures.

Findings

Identified bounds for minimum and maximum leaf status.

Determined bounds for minimum and maximum internal status.

Characterized extremal trees with given diameter or maximum degree.

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 give the smallest and largest values for the minimum leaf status, maximum leaf status, minimum internal status, and maximum internal status of a tree and characterize the extremal cases. We also discuss these parameters of a tree with given diameter or maximum degree.