# On Walks and Canonical Double Coverings of Graphs with the Same Main Eigenspace

The main eigenvalues of a graph $G$ are the eigenvalues of its $(0,1)$-adjacency matrix having some corresponding eigenvector not orthogonal to the all-ones vector $\mathbf j = (1,\dots,1)$. In this dissertation, the relationship between the main eigenvalues of a graph and the number of walks is discussed. The number of walks $N_k$ of length $k$ in the graph $G$ is expressed solely in terms of the main eigenvalues and main angles of $G$. The walk matrices of two comain non-isomorphic graphs with the same main eigenspace are shown to be of the same column space. Moreover, various properties of graphs relating the main eigenvalues, eigenspaces, eigenvectors and canonical double covers are catergorised in a hierarchical form.