Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2601.04050

Planarity criteria for metric graphs

Alice Brolin·Pavel Kurasov
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Colin de Verdi\`ere parameter is a number assigned to discrete graphs which equals the maximal multiplicity of the second eigenvalue of a certain family of Laplacian matrices related to the graph. In this paper it is shown that the Colin de Verdi\`ere parameter can be obtained in the setting of metric graphs by looking at the maximal multiplicity of the second eigenvalue for Laplacians on metric graphs with delta couplings at the vertices. Two different families of Laplacians, as well as a family of Schr\"odinger operators, all leading to the Colin de Verdi\`ere number are presented.

Explore related subjects

math.SPmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alice Brolin, Pavel Kurasov. 2026-01-07. Planarity criteria for metric graphs. https://arxiv.org/abs/2601.04050

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

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗
arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗
arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗