Subdegrees and Suborbital Graphs of Symmetry Groups of Platonic Solids Acting on the Edges of the Respective Solids

Abstract

The action of the symmetry groups of platonic solids on the edges of the respective solids is studied. The symmetry groups of platonic solids have previously been studied by Benson and Grove (1971) and Mokami (2011). In particular, the cycle indices of the action have been investigated. In this research, we employ other methods or otherwise quote these findings for completeness purposes. However, the corresponding ranks, subdegrees and suborbital graphs have not been investigated. Thus, this project deals with the computation of the ranks, subdegrees and the construction of the corresponding suborbital graphs and their properties. The ranks and subdegrees are determined using algebraic concepts such as the Burnside’s lemma and the stabilizer of the edges. The corresponding suborbital graphs are constructed using methods developed by Sims (1967), while the properties are determined using concepts from Graph Theory, such as determining the number of graphs using the rank computed, computing the girth of the graphs and determining whether the graphs are connected or disconnected. In particular, using the rank obtained and the suborbitals corresponding to the action on each solid, the suborbital graphs are constructed and their properties analyzed. The ranks of the tetrahedron, cube, octahedron, dodecahedron and icosahedron have been found to be 4, 7, 7, 16 and 16 respectively. The subdegrees of the tetrahedron have been found to be 1,1,2,2 while those of the cube and octahedron have been found to be 1,1,2,2,2,2,2. The subdegrees of the dodecahedron and icosahedron have been found to be 1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2. It has been found that the number of self-paired graphs of the tetrahedron, cube, octahedron, dodecahedron and icosahedron have been found to be 2, 5, 5, 8 and 8 respectively. The directed suborbital graphs of the tetrahedron, cube and octahedron have a girth of three. Also, based on concepts developed by Sims (1967), the disconnected graphs on all solids show that the action is imprimitive