Abstract. The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural generalisation of the discrete Laplace operator on the underlying graph. These generalised Laplacians are necessary in order to cover general vertex conditions on the metric graph. In case of the standard (also named “Kirchhoff”) conditions, the discrete operator is the usual combinatorial Laplacian. 1.

Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.

We consider the Schrödinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. In example we show the existence of real and complex resonances for some specific potentials. 1 Introduction and main results Consider the Schrödinger operator H = − ∆ + Vq with a periodic potential Vq on so called armchair graph Γ N, N � 1. In order to describe the graph Γ N, we introduce the fundamental cell ˜ Γ = ∪j∈N6 ˜ Γj ⊂ R 2, where ˜ Γj = {x = ˜rj + tej, t ∈ [0, 1]} are edges of length 1, Nm = {1, 2,.., m}, e1 = e6 = 1

Abstract. We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states. 1.

We consider the Schrödinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe the absolutely continuous spectrum of the Schrödinger operator: 1) the multiplicity, 2) endpoints of the gaps, they are given by periodic or antiperiodic eigenvalues or resonances (branch points of the Lyapunov function), 3) resonance gaps, where the Lyapunov function is non-real. We determine the asymptotics of the gaps at high energy. 1 Introduction and main results Consider the Schrödinger operator H = − ∆ + Vq with a periodic potential Vq on so called armchair graph Γ N, N � 1. In order to describe the graph Γ N, we introduce the fundamental cell ˜ Γ = ∪ 6 1 ˜ Γj ⊂ R 2, where ˜ Γj = {x = ˜rj + tej, t ∈ [0, 1]}, j ∈ N6 is the edge of length 1, and Nm = {1, 2,.., m}, e1 = e6 = 1