  We consider a class of diffusion problems defined on simple graphs in which
the populations at any two vertices may be averaged if they are connected by an
edge. The diffusion polytope is the convex hull of the set of population
vectors attainable using finite sequences of these operations. A number of
physical problems have linear programming solutions taking the diffusion
polytope as the feasible region, e.g. the free energy that can be removed from
plasma using waves, so there is a need to describe and enumerate its extreme
points. We review known results for the case of the complete graph $K_n$, and
study a variety of problems for the path graph $P_n$ and the cyclic graph
$C_n$. We describe the different kinds of extreme points that arise, and
identify the diffusion polytope in a number of simple cases. In the case of
increasing initial populations on $P_n$ the diffusion polytope is topologically
an $n$-dimensional hypercube.
