The spectrum of the p-Laplacian and p-harmonic morphisms on graphs

For a real number p with 1 < p < infinity we consider the spectrum of the p-Laplacian on graphs, p-harmonic morphisms between two graphs, and estimates for the solutions of p-Laplace equations on graphs. More precisely, we prove a Cheeger type inequality and a Brooks type inequality for infinite graphs. We also define p-harmonic morphisms and horizontally conformal maps between two graphs and prove that these two concepts are equivalent. Finally, we give some estimates for the solutions of p-Laplace equations, which coincide with Green kernels in the case p = 2.