  In this paper, we examine covering graphs that are obtained from the
$d$-dimensional integer lattice by adding pendant edges. In the case of $d=1$,
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 $d=2$, we show that there exists an arrangement of
the pendant edges such that the Laplacian has no spectral gap.
