On the [formula omitted]-torsional rigidity of combinatorial graphs

Abstract

We study the p-torsion function and the corresponding p-torsional rigidity associated with p-Laplacians and, more generally, p-Schrödinger operators, for 1<p<∞, on possibly infinite combinatorial graphs. We present sufficient criteria for the existence of a summable p-torsion function and we derive several upper and lower bounds for the p-torsional rigidity. Our methods are mostly based on novel surgery principles. As an application, we also find some new estimates on the bottom of the spectrum of the p-Laplacian with Dirichlet conditions, thus complementing some results recently obtained in Mazón and Toledo (2023) in a more general setting. Finally, we prove a Kohler–Jobin inequality for combinatorial graphs (for p=2): to the best of our knowledge, graphs thus become the third ambient where a Kohler–Jobin inequality is known to hold.