Poincar\'e inequalities on graphs

We prove local $L^p$-Poincar\'e inequalities, $ p\in[1,\infty]$, on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global $L^p$-Poincar\'e inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.