Geodesic
The walks and CDC of graphs with the same main eigenspace
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 507-532

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The main eigenvalues of a graph G are those eigenvalues of the (0,1)-adjacency matrix 𝐀 with a corresponding eigenspace not orthogonal to 𝐣 = (1| 1| ⋯ | 1)^𝖳. The principal main eigenvector associated with a main eigenvalue is the orthogonal projection of the corresponding eigenspace onto 𝐣. The main eigenspace of a graph is generated by all the principal main eigenvectors and is the same as the image of the walk matrix. We explore a new concept to see to what extent the main eigenspace determines the entries of the walk matrix of a graph. The CDC of a graph G is the direct product G× K_2. We establish a hierarchy of inclusions connecting classes of graphs in view of their CDC, walk matrix, main eigenvalues and main eigenspaces. We provide a new proof that graphs with the same CDC are characterized as TF-isomorphic graphs. A complete list of TF-isomorphic graphs on at most 8 vertices and their common CDC is also given.
Keywords: bipartite (canonical) double covering, main eigenspace, comain graphs, walk matrix, two-fold isomorphism 
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     title = {The walks and {CDC} of graphs with the same main eigenspace},
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Sciriha, Irene; Collins, Luke. The walks and CDC of graphs with the same main eigenspace. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 507-532. http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a13/