Equilateral quantum graphs and boundary triples

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 boundary conditions on the metric graph. In case of the standard (also named ``Kirchhoff'') boundary conditions, the discrete operator is the usual combinatorial Laplacian.