Related papers: On the hot spots of quantum graphs

It is known that up to certain pathologies, a compact metric graph with standard vertex conditions has a Baire-generic set of choices of edge lengths such that all Laplacian eigenvalues are simple and have eigenfunctions that do not vanish…

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…

This work presents conjectures about eigenvalues of matrices associated with $k$-path graphs, the algebraic connectivity, defined as the second smallest eigenvalue of the Laplacian matrix, and the $\alpha$-index, as the largest eigenvalue…

Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…

We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation…

In this paper we consider particular graphs defined by $\overline{\overline{\overline{K_{\alpha_1}}\cup K_{\alpha_2}}\cup\cdots \cup K_{\alpha_k}}$, where $k$ is even, $K_\alpha$ is a complete graph on $\alpha$ vertices, $\cup$ stands for…

We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking…

We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic…

We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…

There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to…

We analyze graphs attaining the extreme values of various spectral indices in the class of all simple connected graphs, as well as in the class of graphs which are not complete multipartite graphs. We also present results on density of…

We present monotonicity inequalities for certain functions involving eigenvalues of $p$-Laplacians on signed graphs with respect to $p$. Inspired by such monotonicity, we propose new spectrum-based graph invariants, called (variational)…

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the…

To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference…

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…

Consider two quantum graphs with the standard Laplace operator and non-Robin type boundary conditions at all vertices. We show that if their eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then the…

This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from $N$ random samples on a $d$-dimensional manifold embedded in a possibly high dimensional…

Let $\Gamma$ be a locally finite graph, $L$ the normalized Laplacian of $\Gamma$. If $\Gamma$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard…

Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on…