On the complexity of Archimedean solids

The complexity of 13 Archimedean solids via their Schlegel graphs was studied by four indices: the complexity index based on the augmented vertex-degree, and the total numbers of walks, trails and paths. All four descriptors consider the truncated tetrahedron, the truncated cube and the truncated octahedron as the least complex structures, and the rhombicuboctahedron, the icosidodecahedron, the rhombicosidodecahedron, the snub cuboctahedron and the snub icosidodecahedron the most complex structures among the 13 Archimedean solids. The ordering of remaining five Archimedean solids (the truncated icosahedron, the truncated dodecahedron, the cuboctahedron, the truncated cuboctahedron and the truncated icosidodecahedron) differs from index to index. The visualization of the complexity relationship between Archimedean solids is realized by the partial order of their indices in consonance with the Hasse diagram.