On the Walks and Bipartite Double Coverings of Graphs with the same Main Eigenspace

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.