Perfect 2-colorings of inflation of k4

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts A1, . . . , Am such that, for all i, j \in {1,…,m}, every vertex of Ai is adjacent to the same number of vertices, namely, aij vertices, of Aj. The matrix A = (aij) i;j \in {1,2,…,m}, is called the parameter matrix. We study the perfect 2- colorings (also known as the equitable partitions into two parts) of the inflation of k4. In particular, we classify all the realizable parameter matrices of perfect 2-colorings for the inflation of k4.