Nontrivial Eigenvalue Research Articles

Affinity-dependent bound on the spectrum of stochastic matrices

Affinity has proven to be a useful tool for quantifying the non-equilibrium character of time continuous Markov processes since it serves as a measure for the breaking of time reversal symmetry. It has recently been conjectured that the number of coherent oscillations, which is given by the ratio of imaginary and real part of the first non-trivial eigenvalue of the corresponding master matrix, is constrained by the maximum cycle affinity present in the network. In this paper, we conjecture a bound on the whole spectrum of these master matrices that constrains all eigenvalues in a fashion similar to the well known Perron–Frobenius theorem that is valid for any stochastic matrix. As in other studies that are based on affinity-dependent bounds, the limiting process that saturates the bound is given by the asymmetric random walk. For unicyclic networks, we prove that it is not possible to violate the bound by small perturbation of the asymmetric random walk and provide numerical evidence for its validity in randomly generated networks. The results are extended to multicyclic networks, backed up by numerical evidence provided by networks with randomly constructed topology and transition rates.

Eigenvalue-free interval for threshold graphs

This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which −1 and 0 are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval [(−1−2)/2,(−1+2)/2]. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).

Surgery principles for the spectral analysis of quantum graphs

We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet, or $\delta$-type) which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include “transplantation” of volume within a graph based on the behaviour of its eigenfunctions, as well as “unfolding” of local cycles and pendants. In other cases we establish sharp generalisations, extensions, and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions, and introducing new pendant subgraphs. To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest non-trivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates — one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) — and includes them as special cases.

A sharp estimate for Neumann eigenvalues of the Laplace–Beltrami operator for domains in a hemisphere

Here, we prove an isoperimetric inequality for the harmonic mean of the first [Formula: see text] non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of [Formula: see text].

First eigenvalue of the $p$-Laplacian on Kähler manifolds

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on K\"ahler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on K\"ahler manifolds with positive Ricci curvature.

The Steklov Problem on Differential Forms

AbstractIn this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\ldots,v_k$ and each edge $uv$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been investigated as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are 1. a uniform random lift by a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues bounded by $\lambda+\mathcal{O}(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. The probability bounds as well as the dependency on $\lambda$ are almost optimal. As a special case, we obtain that there is a constant $c_1$ such that for every $k\leq 2^{c_1n/d^2}$, there exists a lift $H$ of every Ramanujan graph by a cyclic group of order $k$ such that $H$ is almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This result leads to a quasi-polynomial time deterministic algorithm to construct almost Ramanujan expanders; 2. there is a constant $c_2$ such that for every $k\geq 2^{c_2nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph such that $H$ is almost Ramanujan. This can be viewed as an analogue of the well-known nonexpansion result for constant degree abelian Cayley graphs. Suppose $k_0$ is the order of the largest abelian group that produces expanding lifts. Our two results highlight lower and upper bounds on $k_0$ that are tight up to a factor of $d^3$ in the exponent, thus suggesting a threshold phenomenon.

On Optimality of Second-Highest Policy for Opportunistic Multichannel Access

We consider an opportunistic communication system in which one transmitter communicates with one receiver by one of $N$ two-state Markov channels. The transmitter probes a channel before access. In particular, the transmitter does not access the probed channel if it is found to be in a bad state. Taking into account the probing cost, the transmitter will transmit over the chosen channel for a fixed time interval after probing. To maximize the throughput of the transmitter, we propose the second-highest probing policy, i.e., probing the second-best channel in terms of available probabilities of those channels. Further, we present three sets of conditions to guarantee the optimality of the policy for three scenarios, respectively. The conditions show that the optimality of the policy is tightly coupled with initial belief vector and non-trivial eigenvalue of two-state transition matrix. In addition, we extend the optimality of the policy to two related scenarios from the standpoint of exploitation versus exploration.

Critical Domains for the First Nonzero Neumann Eigenvalue in Riemannian Manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace–Beltrami operator on bounded subdomains $$\Omega $$ of a Riemannian manifold $$({\mathscr {M}},g)$$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $$\mu _2(\Omega )$$ with respect to volume-preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger’s shape derivative formula which covers the case when $$\mu _2(\Omega )$$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $${\mathscr {M}}= {\mathbb {R}}^k \times {\mathscr {N}}$$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.

Sharp Poincaré inequalities in a class of non-convex sets

Let \gamma be a smooth, non-closed, simple curve whose image is symmetric with respect to the y -axis, and let D be a planar domain consisting of the points on one side of \gamma , within a suitable distance \delta of \gamma . Denote by \mu_1^{\textup{odd}}(D) the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the y -axis. If \gamma satisfies some simple geometric conditions, then \mu_1^{\mathrm{odd}}(D) can be sharply estimated from below in terms of the length of \gamma , its curvature, and \delta . Moreover, we give explicit conditions on \delta that ensure \mu_1^{\mathrm{odd}}(D)=\mu_1(D) . Finally, we can extend our bound on \mu_1^{\mathrm{odd}}(D) to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. Lower estimates of the first non-trivial Neumann eigenvalues of the p-Laplace operator in cusp domains $$\Omega \subset \mathbb R^n$$ , $$n\ge 2$$ , are given.

On the First Eigenvalue of the Degenerate $$\varvec{p}$$ p -Laplace Operator in Non-convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$ , in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincare–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs

We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation , random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for $(n,d,\lambda)$ -graphs (i.e., $d$ -regular graphs on $n$ vertices with all non-trivial eigenvalues of the adjacency matrix less than $\lambda$ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability $p\geq1-C_{1}n^{-C_{2}/d}$ with $C_{1}$ and $C_{2}$ depending only on $\lambda /d$ .

Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian

In this paper we study integral estimates of derivatives of conformal mappings φ:D→Ω of the unit disc D⊂C onto bounded domains Ω that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev–Poincaré inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann–Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann–Laplace operator in some fractal type domains (snowflakes) were obtained.

Curious Convergence Properties of Lattice Boltzmann Schemes for Diffusion with Acoustic Scaling

We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above.

A toy Neumann analogue of the nodal line conjecture

We introduce an analogue of Payne’s nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain $$\Omega $$ with rotational symmetry of order two (i.e. $$x\in \Omega $$ iff $$-x\in \Omega $$ ) “should normally” be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.

Orbit equivalence of Cantor minimal systems and their continuous spectra

To any continuous eigenvalue of a Cantor minimal system $$(X,\,T)$$ , we associate an element of the dimension group $$K^0(X,\,T)$$ associated to $$(X,\,T)$$ . We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $$(G,\,u)$$ (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs

Following Escobar (J Funct Anal 150(2):544–556, 1997) and Jammes (Ann l’Inst Fourier 65(3):1381–1385, 2015), we introduce two types of isoperimetric constants and give lower bound estimates for the first nontrivial eigenvalues of Dirichlet-to-Neumann operators on finite graphs with boundary respectively.

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

Edge connectivity and the spectral gap of combinatorial and quantum graphs

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds corresponds to a well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers.Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are ‘asymptotically correct’, i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.