On the $p$-torsional rigidity of combinatorial graphs

Abstract

We study the $p$-\emph{torsion function} and the corresponding $p$-\emph{torsional rigidity} associated with $p$-Laplacians and, more generally, $p$-Schr\"odinger operators, for $1<p<\infty$, 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: Jos\'e M. Maz\'on, Julian Toledo, Torsional rigidity in random walk spaces, 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.