  We study the $p$-independence of spectra of Laplace operators on graphs
arising from regular Dirichlet forms on discrete spaces. Here, a sufficient
criterion is given solely by a uniform subexponential growth condition.
Moreover, under a mild assumption on the measure we show a one-sided spectral
inclusion without any further assumptions. We study applications to normalized
Laplacians including symmetries of the spectrum and a characterization for
positivity of the Cheeger constant. Furthermore, we consider Laplacians on
planar tessellations for which we relate the spectral $p$-independence to
assumptions on the curvature.
