  The main eigenvalues of a graph $G$ are those eigenvalues of the
$(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not
orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct
product $G\times K_2$. The main eigenspace of $\mathbf A$ is generated by the
principal main eigenvectors and is the same as the image of the walk matrix. A
hierarchy of properties of pairs of graphs is established in view of their
CDC's, walk matrices, main eigenvalues, eigenvectors and eigenspaces. We
determine by algorithm that there are 32 pairs of non-isomorphic graphs on at
most 8 vertices which have the same CDC.
