Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.38670

Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices

Hongjin Wu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study how the structure of the coupling matrix $\boldsymbol A$ shapes the energy landscape $E_{\boldsymbol A}(\boldsymbolθ)=-\frac12\sum_{i,j}A_{ij}\cos(θ_i-θ_j)$ of the Kuramoto model. We prove that, for an entrywise nonnegative coupling matrix $\boldsymbol A$ with connected support, if $\boldsymbol A$ has rank-width one over $\mathbb R$, then the landscape is "simple", i.e., every local minimum is synchronized. The rank-width bound is sharp, and the result has many interesting implications for global synchronization. The key contribution of this work is to uncover a structural synchronization mechanism beyond the rank-one regime, which has served as a starting point for earlier analyses of global synchronization.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hongjin Wu. 2026-10-28. Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices. https://arxiv.org/abs/2610.38670

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How to Tropicalize a non-Archimedean Lattice

We present a method for computing the tropicalization of a lattice over the valuation ring of a non-Archimedean field. Our construction relies on the entropy polynomial, a tropical polynomial whose coefficients are determined by the valuations of minors of a matrix representing the lattice. We show that the tropicalization of the lattice is the support of a polyhedral complex defined through this polynomial. Moreover, we prove that this tropicalization is a finitely generated tropical semimodule over the nonnegative tropical scalars and provide explicit generators in terms of the entropy vector of the lattice. Finally, we extend our construction to valuated bimatroids, establishing analogous results beyond the realizable setting.

math.CO↗
arXiv2026

Arboreal Networks and Ultrametrics

Ultrametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

math.CO↗
arXiv2026

Coloring t-perfect graphs with fewer colors

Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 130 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction to t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.

math.CO↗