Spectra of semi-regular polytopes

Spectra of semi-regular polytopes

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

Spectra of Regular Polytopes

Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of...

On the Size of Equifacetted Semi-regular Polytopes

Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group.

Regular polytopes

Radii of Regular Polytopes

There are three types of regular polytopes which exist in every dimension d: regular simplices, (hyper-) cubes, and regular cross-polytopes. In this paper we investigate two pairs of inner and outer j-radii (rj, Rj) and (r̄j, R̄j) of these polytopes (inner and outer radii classes are almost always considered in pairs, such that for a 0-symmetric body K and its dual K the inner (outer) radii of K ...