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

TL;DR

This paper establishes a condition on weighted graphs with finite measure that ensures a global Poincaré inequality, drawing parallels with classical criteria in geometric analysis.

Contribution

It introduces a new discrete analogue of classical geometric criteria, linking weighted graph properties to Poincaré inequalities.

Findings

Condition guarantees global Poincaré inequality

Links discrete graph conditions to classical geometric criteria

Extends classical analysis to weighted graphs

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.