Before revision: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality . This is analogous to the Cheeger-type inequality relating the spectral gap  , \lambda_2 (G), of the graph to its {\em edge expansion}.
After revision: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality , analogous to the Cheeger-type inequality relating the spectral gap , \lambda_2 (G), of the graph to its {\em edge expansion}.