Prescribed-time fully distributed optimization for time-varying costs: Zero-gradient-sum scheme.

Improved upper bound of multiplicity of (signless) Laplacian eigenvalue two

Reconstruction of Schrödinger operators by half of the Dirichlet eigenvalues

We introduce and study the logarithmic p p -Laplacian L Δ p L_{\Delta _p} , which emerges from the formal derivative of the fractional p p -Laplacian ( − Δ p ) s (-\Delta _p)^s at s = 0 s=0 . This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator (see H. Chen and T. Weth [Comm. Partial Differential Equations 44 (2019), pp. 1100–1139]). We present a variational framework to study the Dirichlet problems involving the L Δ p L_{\Delta _p} in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional p p -Laplacian and the logarithmic p p -Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of L Δ p L_{\Delta _p} . We discuss maximum and comparison principles for L Δ p L_{\Delta _p} in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of L Δ p L_{\Delta _p} . In addition, we prove that the first Dirichlet eigenfunction of L Δ p L_{\Delta _p} is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic p p -Laplacian.

Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds

A bstract In recent years the conformal bootstrap has produced surprisingly tight bounds on many non-perturbative CFTs. It is an open question whether such bounds are indeed saturated by these CFTs. A toy version of this question appears in a recent application of the conformal bootstrap to hyperbolic orbifolds, where one finds bounds on Laplace eigenvalues that are exceptionally close to saturation by explicit orbifolds. In some instances, the bounds agree with the actual values to 11 significant digits. In this work we show, under reasonable assumptions about the convergence of numerics, that these bounds are not in fact saturated. In doing so, we find formulas for the OPE coefficients of hyperbolic orbifolds, using links between them and the Rankin-Cohen brackets of modular forms.

Abstract In this paper, we study the so-called clamped transmission eigenvalue problem. This is a new transmission eigenvalue problem that is derived from the scattering of an impenetrable clamped obstacle in a thin elastic plate. The scattering problem is modeled by a biharmonic wave operator given by the Kirchhoff–Love infinite plate problem in the frequency domain. These scattering problems have not been studied to the extent of other models. Unlike other transmission eigenvalue problems, the problem studied here is a system of homogeneous PDEs defined in all of R 2 . This provides unique analytical and computational difficulties when studying the clamped transmission eigenvalue problem. We are able to prove that there exist infinitely many real clamped transmission eigenvalues. This is done by studying the equivalent variational formulation. We also investigate the relationship of the clamped transmission eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian for the bounded scattering obstacle.

Shape derivative of the Laplacian eigenvalue problem

Motivated by the need to distinguish connectivity from robustness and to enable comparability across networks of varying scales, this work introduces a new measure termed the spectral connectivity extent . is defined as the average of the second and third smallest Laplacian eigenvalues, normalized by network size and rescaled to the interval [0, 100]. Higher values of correspond to stronger network connectivity. Several networks are analyzed to evaluate the proposed measure. By analyzing a real-world network, it is found that when the network undergoes a critical change, the relative variation of and the global efficiency are 16.485 and 0.037, respectively, whereas under an unimportant change, the relative variations of and are 0.2 and 0.162, respectively. These findings indicate that, compared with existing measures, more effectively captures significant structural changes within the network. Since emphasizes the efficiency of interactions among nodes, it has potential applications in the design of communication and transportation networks, the analysis of disease spreading, and the assessment of infrastructure resilience.

Abstract Given a conformal action of a discrete group on a Riemann surface, we study the maximization of Laplace and Steklov eigenvalues within a conformal class, considering metrics invariant under the group action. We establish natural conditions for the existence and regularity of maximizers. Our method simplifies the previously known techniques for proving existence and regularity results in conformal class optimization. Finally, we provide a complete solution to the equivariant maximization problem for Laplace eigenvalues on the sphere and Steklov eigenvalues on the disk, resolving open questions posed by Arias‐Marco et al. (2024) regarding the sharpness of the Hersch–Payne–Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk among planar simply connected domains with symmetry.

Upper Bounds on the Second Largest Laplacian Eigenvalue of Graphs with Given Size

For a graph $G$ of $n$ vertices, let $\mu_{1}(G)$ be its largest Laplacian eigenvalue. It was conjectured by Ashraf et al. in [Electron. J. Combin. 21(3):#P3.6 (2014) that$$ \mu_{1}(G) \mu_{1}(\bar{G}) \leqslant n(n-1), $$where $\bar{G}$ is the complement of $G$, and equality holds if and only if $G$ or $\bar{G}$ is isomorphic to the join of an isolated vertex and a disconnected graph of order $n-1$. They proved that this conjecture holds for bipartite graphs. In this paper, we completely confirm this conjecture. Furthermore, we propose a more general conjecture that for any graph $G$ with $n$ vertices and $k \leq \frac{3n}{4}$,$$ \mu_k(G) \mu_k(\bar{G})\leq n(n-k), $$and equality holds if and only if $G$ or $\bar{G}$ is isomorphic to the join of $K_{k}$ and a disconnected graph on $n-k$ vertices and has at least $k+1$ connected components. We also prove that it is true for $\frac{n}{2}\leq k \leq \frac{3n}{4}$, and for each $k \geq \frac{3n}{4}+1$, a counterexample is given.

Sharp upper bounds on the second largest signless Laplacian eigenvalues of connected graphs

Laplacian eigenvalue distribution in terms of degree sequence

Upper bound of the multiplicity of Laplacian eigenvalue 1 of trees

ABSTRACT In recent years, time‐varying formations have emerged as a crucial control strategy due to their distinct advantages in adapting to environmental changes. However, achieving time‐varying formation tracking in heterogeneous multi‐agent systems via reinforcement learning (RL), especially when the leader's state is unavailable, remains a significant challenge. This paper investigates the problem of time‐varying formation tracking for heterogeneous multi‐agent systems (MASs) with unknown dynamics and inaccessible leader states. A fully distributed output‐feedback reinforcement learning framework is developed, which integrates adaptive observer design, optimal regulation, and data‐driven policy iteration into a unified control scheme. Specifically, a fully distributed adaptive observer is proposed to estimate the leader's state using only its output, without requiring Laplacian eigenvalues or global network knowledge. Based on this observer, an output‐feedback reinforcement learning controller is constructed to achieve asymptotic convergence of the formation tracking error, in contrast to existing state‐feedback‐based methods that only ensure bounded errors. Furthermore, a state reconstruction mechanism, originally used in synchronization problems, is extended to time‐varying formation tracking, enabling policy learning directly from input‐output data under unknown dynamics. Theoretical analysis and simulation studies demonstrate that the proposed framework achieves robust, scalable, and model‐free time‐varying formation tracking, offering clear advantages over existing approaches.

Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend the ideas of Rohleder to a new proof of Friedlander’s inequality for any bounded Lipschitz domain.

Abstract We perform a systematic variational method for functionals depending on eigenvalues of Riemannian manifolds. It is based on a new concept of Palais–Smale (PS) sequences that can be constructed thanks to a generalization of classical min‐max methods on functionals to locally Lipschitz functionals. We prove convergence results on these PS sequences emerging from combinations of Laplace eigenvalues or combinations of Steklov eigenvalues in dimension 2.

Let G be a connected graph with vertex set V ( G ) and edge set E ( G ) . The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is a diagonal matrix of degrees of the vertices of G and A ( G ) is the adjacency matrix of G. The multiplicity of an eigenvalue μ of L ( G ) is denoted by m G ( μ ) . In 2022, Wen et al. [Czech. Math. J., 72(2022)] proved that, if G is not a cycle, then m G ( μ ) ≤ 2 c ( G ) + p ( G ) − 1 , where c ( G ) = | E ( G ) | − | V ( G ) | + 1 is the cyclomatic number of G and p ( G ) is the number of pendant vertices of G. We characterize the graph G with m G ( 1 ) = 2 c ( G ) + p ( G ) − 1 .