Perfect 2-colorings of the Johnson graph J(9, 4)

Abstract

A perfect 2-coloring of a graph $$\varGamma$$ with matrix $$M=\{m_{ij}\}_{i, j=1, 2}$$ is a coloring of the vertices $$\varGamma$$ with colors called $$\{1, 2\}$$ such that the number of vertices of color j adjacent to a fixed vertex of color i is equal to $$m_{ij}$$ . We state the matrix M is the parameter matrix. Each class of an equitable partition is the vertices with the same color. In this article, we classify the parameter matrices of whole perfect 2-colorings of the Johnson graph J(9, 4).