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 .
After revision:  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 .