  Property A is a form of weak amenability for groups and metric spaces
introduced as an approach to the famous Novikov higher signature conjecture,
one of the most important unsolved problems in topology.
  We show that property A can be reduced to a sequence of linear programming
optimization problems on finite graphs. We explore the dual problems, which
turn out to have interesting interpretations as combinatorial problems
concerning the maximum total supply of flows on a network.
  Using isoperimetric inequalities, we relate the dual problems to the Cheeger
constant of the graph and explore the role played by symmetry of a graph to
obtain a striking characterization of the difference between an expander and a
graph without property A. Property A turns out to be a new measure of
connectivity of a graph that is relevant to graph theory.
  The dual linear problems can be solved using a variety of methods, which we
demonstrate on several enlightening examples. As a demonstration of the power
of this linear programming approach we give elegant proofs of theorems of Nowak
and Willett about graphs without property A.
