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

Abstract

For a real number $p$ with $1<p<\infty$ 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 $p=2$.