We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincaré Anal.\ Non Linéaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions.
We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions associated with the nodal sets as the nodal domains of Courant-sharp eigenfunctions.
We consider spectral minimal partitions. Continuing work of the the present authors about problems for planar domains, [23], we focus on the sphere and obtain a sharp result for 3-partitions which is related to questions from harmonic analysis, in particular to a conjecture of Bishop.
Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides…
We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Δ+ V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Σ$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $k\in\…
We show that a recent spectral flow approach proposed by Berkolaiko-Cox-Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction can be extended to more general partitions. To be more precise, we work with spectral equipartitions that satisfy a pair compatible condition. Nodal partitions and spectral minimal partitions are examples of such partitions. Along the way, we discuss different approaches to the Dirichlet-to-Neumann operators: via Aharonov-Bohm operators, via a double covering argument, and via a slitting of the domain.
In this survey, we review the properties of minimal spectral $k$-partitions in the two-dimensional case and revisit their connections with Pleijel's Theorem. We focus on the large $k$ problem (and the hexagonal conjecture) in connection with two recent preprints by J. Bourgain and S. Steinerberger on the Pleijel Theorem. This leads us also to discuss some conjecture by I. Polterovich, in relation with square tilings. We also establish a Pleijel Theorem for Aharonov-Bohm Hamiltonians and deduce from it, via the magnetic characterization of the minimal partitions, some lower bound for the number…
We study a minimal partition problem on the flat rectangular torus. We give a partial review of the existing literature, and present some numerical and theoretical work recently published elsewhere by V. Bonnaillie-No{ë}l and the author, with some improvements.
We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one…
Given a bounded open set $Ω$ in $ \mathbb R^n$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_j$, we consider the quantity $\max_j λ(D_j)$ where $λ(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $ \mathfrak L_k(Ω)$ the infimum over all the $k$-partitions of $ \max_j λ(D_j)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$'s is non trivial and quite interesting. In this paper, we give the p…
Given a bounded open set $Ω$ in $\mathbb{R}^n$ (or a compact Riemannian manifold with boundary), and a partition of $Ω$ by $k$ open sets $ω_j$, we consider the quantity $\max_j λ(ω_j)$, where $λ(ω_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $ω_j$. We denote by $\mathfrak{L}_k(Ω)$ the infimum of $\max_j λ(ω_j)$ over all $k$-partitions. A minimal $k$-partition is a partition which realizes the infimum. The purpose of this paper is to revisit properties of nodal sets and to explore if they are also true for minimal partitions, or more generally for spectral equ…
We study partitions of the rectangular two-dimensional flat torus of length 1 and width b into k domains, with b a parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, definedas the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are inparticular interested in the way these minimal partitions change when b is varied. We present herean improvement, when k is odd, of the results on transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establis…
An asymptotic formula for the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold is given under a integrality assumption on the spectrum of the Hamiltonian. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer.
For random graphs distributed according to a stochastic block model, we consider the inferential task of partioning vertices into blocks using spectral techniques. Spectral partioning using the normalized Laplacian and the adjacency matrix have both been shown to be consistent as the number of vertices tend to infinity. Importantly, both procedures require that the number of blocks and the rank of the communication probability matrix are known, even as the rest of the parameters may be unknown. In this article, we prove that the (suitably modified) adjacency-spectral partitioning procedure, re…
This survey is a short version of a chapter written by the first two authors in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin: De Gruyter, 2017] (where more details and references are given) but we have decided here to put more emphasis on the role of the Aharonov-Bohm operators which appear to be a useful tool coming from physics for understanding a problem motivated either by spectral geometry or dynamics of population. Similar questions appear also in Bose-Einstein theory. Finally some open problems which might be of interest are mentioned.
We analyze spectral minimal $k$-partitions for the torus. In continuation with what we have obtained for thin annuli or thin strips on a cylinder (Neumann case), we get similar results for anisotropic tori.
The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equ…
On spectral minimal partitions II, the case of the rectangle
IRMAR, ENS Cachan Betagne, Univ. Rennes I, CNRS（雷恩第一大学，法国国家科学研究中心）；Département de Mathématiques, Université Paris-Sud（巴黎第十一大学数学系）；Institut für Theoretische Chemie, Universität Wien and International Erwin Schrödinger Institute for Mathematical Physics（维也纳大学理论化学研究所，国际埃尔温·薛定谔数学物理研究所）
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities, i.e.…
We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every $n$-vertex graph of genus $g$ and maximum degree $d$ satisfies $λ_2(G) = O((g+1)^3 d/n)$. This recovers the $O(d/n)$ bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of $O((g+1) poly(d)/n)$, but our proof does not make use of conformal mappings or circle packings. We are thu…
