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

In this paper, we examine covering graphs that are obtained from the d  -dimensional integer lattice by adding pendant edges. In the case of d=1$d=1$, we show that the Laplacian on the graph has a spectral gap and establish a necessary and sufficient condition under which the Laplacian has no eigenvalues. In the case of d=2$d=2$, we show that there exists an arrangement of the pendant edges such that the Laplacian has no spectral gap.