Weighted graphs in the sense of John and a global Poincar\'e inequality

Abstract

In this paper, we establish a condition on weighted graphs with finite measure that guarantees the validity of a global Poincar\'e inequality. This condition can be viewed as a discrete analogue of the criterion introduced by J. Boman in 1982 for Whitney cubes, which in turn characterizes the condition originally proposed by F. John in his seminal 1961 work.