The Wiener index of vertex colorings

TL;DR

This work extends the Wiener index from vertex sets to vertex colorings, studying how to maximize the sum of pairwise distances within color classes. It provides a complete characterization of $k$-color weak and local weak maximizers on paths via canonical families $\mathcal{C}_t$ and a majorization framework, linking color-class sizes to $W$ through the type $t=(n_1,\ldots,n_k)$ with a precise ordering. On cycles, the authors show that the path-like majorization picture fails in general (with explicit counterexamples) and develop refined criteria using the $\mathcal{R}^\infty$-closure and related theorems to describe when $W$ is preserved or ordered among weak maximizers. The paper thus blends combinatorial constructions (good sets, equitable partitions), majorization theory, and distance-based invariants to advance understanding of Wiener-index-driven colorings, while outlining several open problems for broader graph classes and energy notions.

Abstract

The Wiener index of a vertex coloring of a graph is defined to be the sum of all pairwise geodesic distances between vertices of the same color. We provide characterizations of vertex colorings of paths and cycles whose Wiener index is as large as possible over various natural collections. Along the way we establish a connection between the majorization order on tuples of integers and the Wiener index of vertex colorings on paths and cycles.

Figures (8)

• Figure 1: The $3$-color weak maximizers of $W$ on $C_7$ up to automorphism.
• Figure 2:
• Figure 3: Two $3$-color weak maximizers on $C_6$.
• Figure 4: Distances from $\hat{v}$ to vertices of color $j$ and $j'$ when $n$ is even and $n_j=n_{j'}=2a$, where $a\in\mathbb{Z}$.
• Figure 5: Distances from $\hat{v}$ to vertices of color $j$ and $j'$ when $n$ is odd.

Theorems & Definitions (38)

• Definition 1.1
• Definition 2.1
• Theorem 2.2: Bushaw, Cody and Leffler BCL
• Definition 2.3
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof