Related papers: Always detectable eigenfunctions on metric graphs

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…

We study the limit of first eigenfunctions of (discrete) $p$-Laplacian on a finite subset of a graph with Dirichlet boundary condition, as $p\to 1.$ We prove that up to a subsequence, they converge to a summation of characteristic functions…

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result…

A set of vertices $S$ \emph{resolves} a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of a graph $G$ is the minimum cardinality of a resolving set. In this…

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014.…

We introduce the the fractional Laplacian on a subgraph of a graph with Dirichlet boundary condition. For a lattice graph, we prove the upper and lower estimates for the sum of the first $k$ Dirichlet eigenvalues of the fractional…

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…

We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature of the…

The maximal graph Dirichlet problem asks whether there exists a spacelike graph, in a semi-Euclidean space, with a given boundary and with mean curvature everywhere zero. We prove the existence of solutions to this problem under certain…

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that generically (in the…

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of…

We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the…

A theorem of Strichartz states that if a uniformly bounded bi-infinite sequence of functions on the Euclidean spaces, satisfies the condition that the Laplacian acting on a function in this sequence yields the next one, then each function…

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $\Omega\subset K$ whose boundary $\partial \Omega$ is a graph-directed self-similar set, we prove that…

We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds…

An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{\"o}dinger…

We consider the Dirichlet eigenvalues of the fractional Laplacian $(-\Delta)^s$, with $s\in (0,1)$, related to a smooth bounded domain $\Omega$. We prove that there exists an arbitrarily small perturbation $\tilde\Omega=(I+\psi)(\Omega)$ of…

We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The…

In this paper, we obtain some comparisons of the Dirichlet, Neumann and Laplacian eigenvalues on graphs. We also discuss their rigidities and some of their applications including some Lichnerowicz-type, Fiedler-type and Friedman-type…

It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence…