A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

Tsvi Tlusty

Home/Archive/arXiv

arXiv · 1007.4132

A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

Tsvi Tlusty

View original source ↗Read source document ↗Export metadata ↓

Abstract

This note discusses a relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show that the m-dimensional eigenspace of λ2 is tight and thus defines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mapping is shown to set Colin de Verdière's upper bound on the maximal λ2-multiplicity, where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.

Explore related subjects

math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tsvi Tlusty. 2010-07-23. A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces. https://arxiv.org/abs/1007.4132

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

KEEP EXPLORING

Related papers

arXiv2026

A Functorial Theory of Defects in Abelian Chern-Simons Theory

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

math-ph↗

arXiv2026

Gradient nature of Laplacian growth

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a ``gradient descent'' in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

math-ph↗

arXiv2026

Entanglement-Inducing Quantum Markov Processes

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

math-ph↗