Dirichlet-Poincar\'{e} profiles of graphs and groups

We define Poincar'{e} profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincar'{e} profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases $p=1$ and $p=\mathrm{\infty }$, they detect the F\o lner function and the growth function respectively.