[Paper Review] Minimizing Wiener Index for Vertex-Weighted Trees with Given Weight and Degree Sequences

Mikhail Goubko|arXiv (Cornell University)|Feb 4, 2015

Graph theory and applications14 references17 citations

TL;DR

This paper establishes that the vertex-weighted Wiener index (VWWI) of a tree is minimized by a Huffman tree constructed via a bottom-up algorithm that repeatedly merges the two vertices of least weight. The optimal structure is shown to be a greedy tree where internal nodes are formed by combining the smallest-weight subtrees, generalizing classic results on unweighted trees and extending Huffman's algorithm to vertex-weighted graph optimization with fixed degree and weight sequences.

ABSTRACT

In 1997 Klavžar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.

Motivation & Objective

• To identify the tree structure that minimizes the vertex-weighted Wiener index (VWWI) under fixed vertex weights and degree sequences.
• To generalize the classical Huffman algorithm for optimal prefix codes to the domain of vertex-weighted trees.
• To prove that the optimal tree is a 'proper' directed tree where the weight distribution of subtrees satisfies a majorization condition.
• To establish a connection between VWWI minimization and majorization of subtree weight vectors, extending prior results on unweighted trees.
• To provide a constructive algorithm for generating the optimal tree using a bottom-up merging process based on vertex weights.

Proposed method

• Define the vertex-weighted Wiener index (VWWI) as $\frac{1}{2}\sum_{u,v\in V(G)}\mu_G(u)\mu_G(v)d_G(u,v)$, where $\mu_G(v)$ is the vertex weight.
• Introduce the concept of a 'generating tuple' $\langle\mu, d\rangle$ with fixed vertex weights $\mu(v)$ and degrees $d(v)$, satisfying $\sum d(v) = 2(|V|-1)$.
• Construct a directed tree representation to simplify analysis, where each internal node represents a subtree and stores the total weight of its descendants.
• Define the 'vector of subordinate groups' weights $\mathbf{f}$, representing the total weight of subtrees rooted at each internal node.
• Use majorization theory to compare weight distributions: if $\mathbf{f}(H) \succ^w \mathbf{f}(T)$, then $VWWI(H) \leq VWWI(T)$ under concave, increasing functions.
• Apply a transformation to show that any non-proper tree can be improved, proving that the optimal tree must be 'proper' and thus corresponds to a Huffman tree.

Experimental results

Research questions

• RQ1 What tree structure minimizes the vertex-weighted Wiener index when vertex weights and degrees are fixed?
• RQ2 Can the classic Huffman algorithm be adapted to minimize VWWI in vertex-weighted trees with given degree and weight sequences?
• RQ3 How does the majorization of subtree weight vectors relate to VWWI minimization in such trees?
• RQ4 Under what conditions is the greedy tree (built top-down) equivalent to the Huffman tree (built bottom-up) in minimizing VWWI?
• RQ5 What structural properties must an optimal tree satisfy to minimize the VWWI under the given constraints?

Key findings

• The optimal tree minimizing the vertex-weighted Wiener index is the Huffman tree constructed by repeatedly merging the two vertices of smallest weight.
• The optimal tree is a 'proper' tree, meaning no internal node has a subtree weight exceeding $\bar{\mu}/2$, where $\bar{\mu}$ is the total vertex weight.
• The vector of subtree weights in the optimal tree majorizes that of any other tree, ensuring minimal VWWI under the given constraints.
• The VWWI minimization problem reduces to a majorization problem over the weight vectors of subordinate groups, with the optimal solution achieved when the weight vector is as 'balanced' as possible.
• The result generalizes previous findings on unweighted trees: when all vertex weights are equal, the optimal tree coincides with the greedy tree (Volkmann tree) for fixed degree sequences.
• The optimal tree is unique up to isomorphism when the weight sequence is strictly positive and distinct, and the algorithm produces a unique Huffman tree via the bottom-up merging process.