Extremal trees of given degree sequence or segment sequence with respect to Steiner 3-eccentricity

TL;DR

This paper determines extremal trees with given degree or segment sequences that maximize Steiner 3-eccentricity, providing sharp bounds and characterizations for various classes of trees.

Contribution

It establishes the maximum average Steiner 3-eccentricity for trees with specified degree and segment sequences and characterizes the extremal trees achieving these bounds.

Findings

Identified the sharp upper bounds on average Steiner 3-eccentricity for trees with given degree sequences.

Characterized the unique extremal trees with maximum Steiner 3-eccentricity among various classes.

Provided explicit structures of extremal trees based on degree and segment constraints.

Abstract

The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of all $n$-vertex trees, $\mathbb{T}_{n,\Delta}$ be the set of $n$-vertex trees with given maximum degree equal to $\Delta$, $\mathbb{T}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree, and let $\mathbb{T}_{n,\Delta}^k$ be the set of $n$-vertex trees with exactly $k$ vertices of given maximum degree equal to $\Delta.$ In this paper, we first determine the sharp upper bound on the average Steiner 3-eccentricity of $n$-vertex trees with given degree sequence. The corresponding extremal graph is characterized. Consequently, together with majorization theory, the unique graph among $\mathbb{T}_n$ (resp. $\mathbb{T}_{n,\Delta}$, $\mathbb{T}_n^k, \mathbb{T}_{n,\Delta}^k$) having the maximum average Steiner 3-eccentricity is identified. Then we characterize the unique $n$-vertex tree with given segment sequence having the largest average Steiner 3-eccentricity. Similarly, the $n$-vertex tree with given number of segments having the largest average Steiner 3-eccentricity is determined.