Characterizations of the dimension for classes of concordant polytopes

The spectral method is used to completely describe the set of possible dimensions for concordant polytopes arising from complete undirected and directed graphs. For concordant polytopes satisfying c-factor or c-difactor constraints, we determine the more limited sets of possible dimensions and characterize the polytopes of each dimension. We also demonstrate how the spectral method may be used to refine the direct method for determining the dimension of a concordant polytope.