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We address the Laplacian on a perturbed periodic graph which might not be aperiodic graph. We present a class of perturbed graphs for which the essentialspectra of the Laplacians are stable even when the graphs are perturbed byadding and removing infinitely many vertices and edges. Using this result, wedemonstrate how to determine the spectra of cone-like graphs, the upper-halfplane, and graphs obtained from $\mathbb{Z}^2$ by randomly adding vertices.

Essential spectrum of the discrete Laplacian on a perturbed periodic graph