Perfect 3-colorings of Cubic Graphs of Order 8

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect mm-coloring of a graph GG with mm colors is a partition of the vertex set of GG into m parts A1A1, ……, AmAm such that, for all i,j∈{1,⋯,m}i,j∈{1,⋯,m}, every vertex of AiAi is adjacent to the same number of vertices, namely, aijaij vertices, of AjAj . The matrix A=(aij)i,j∈{1,⋯,m}A=(aij)i,j∈{1,⋯,m} is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the cubic graphs of order 88. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order 8.