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We study the $p$-\emph{torsion function} and the corresponding$p$-\emph{torsional rigidity} associated with $p$-Laplacians and, moregenerally, $p$-Schr\"odinger operators, for $1<p<\infty$, on possibly infinitecombinatorial graphs of finite measure. We present sufficient criteria for theexistence of a summable $p$-torsion function and we derive several upper andlower bounds for the $p$-torsional rigidity. Our methods are mostly based onnovel surgery principles. As an application, we also find some new estimates onthe bottom of the spectrum of the $p$-Laplacian with Dirichlet conditions, thuscomplementing some results recently obtained in: Jos\'e M. Maz\'on, JulianToledo, Torsional rigidity in random walk spaces, in a more general setting.Finally, we prove a Kohler-Jobin inequality for combinatorial graphs.

On the $p$-torsional rigidity of combinatorial graphs of finite measure