Spectrum of the Laplacian on a covering graph with pendant edges. I: The one-dimensional lattice and beyond (Q2435551)

This paper deals with the spectrum of the Laplacian on a covering graph obtained by adding pendant edges to the \(d\)-dimensional integer lattice. The results in the paper are mainly on the cases when \(d=1\) or \(d=2\). In the case of \(d=1\), the author shows that the Laplacian has a spectral gap and gives a classification of the Laplacian with no eigenvalues. However, this result does not hold for the case of \(d=2\). To see this, an arrangement of the pendant edges is provided such that the corresponding Laplacian has no spectral gap.