On extreme points of the diffusion polytope

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.