Abstract
Thespectrum spec( ) of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of . The number of non-zeros in spec( ) exceeds the dimension of 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 (see Section 6). (ii) The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6). (iii) The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7). (iv) The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3q vertices (see Section 8).
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