The spectrum spec(𝒫) of a convex polytope 𝒫⊆ℝE is defined as the order (non-increasing) list of squared singular values of [A•1], where the rows of A are the extreme points of 𝒫. The number of non-zeros in spec(𝒫) exceeds the dimension of 𝒫 by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of: (i) The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2q vertices. (ii) The edge-incidence vectors of Hamiltonian tours in the complete undirected graph on n vertices. (iii) The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph of n nodes. (iv) The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3q vertices. In the cases of (ii) and (iii), the associated dimension results are well-known. The dimension results for (i) and (iv) do not seem to be well-known. General principles are discussed for ‘balanced polytopes’ arising from complete structures.
