Related papers: Sharp diameter bound on the spectral gap for quant…

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum…

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum…

We investigate properties of a particle confined to a hard-wall spiral-shaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the…

We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported, we show that the S-matrix for all energies in any open set in the continuous…

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on $n$ vertices with minimal degree linear in $n$ is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that…

Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is…

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal…

We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrodinger equation on…

It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the…

To a unitary matrix U we associate a doubly stochastic matrix M by taking the modulus squared of each element of U. To study the connection between onset of quantum chaos on graphs and ergodicity of the underlying Markov chain, specified by…

This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of…

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a…

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an…

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of…

We briefly comment on the quantum area spectra of black holes, paying particular attention to the size of the spacing between adjacent spectral levels. It has previously been conjectured that this spacing is uniform with a universal value…

Let G be a graph of n vertices and m edges, and let G has no cycles of length 4. We give upper bounds on the adjacency spectral radius of G in terms of n and m.

We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or delta type) boundary conditions with continuous wavefunctions, we investigate two…

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or…