On the walks, coverings, symmetry and conductivity of graphs with the same main eigenspace

The canonical double cover (CDC) of a graph G is the direct product G × K₂. Graphs with the same CDC share the same walk matrix but not necessarily the same main eigenvalues or eigenvectors that determine the number of walks between pairs of vertices. We explore a new concept to see to what extent the main eigenspace determines the entries of the walk matrix of a graph. 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 have two–fold symmetry and are characterized as TF-isomorphic graphs. In the source and sink potential (SSP) model, current flowing through the bonds of a Pi system molecule, from the source atom to the sink one, may choose a shortest path or may take a longer route, possibly flowing along the edges of cycles. Molecular electronic devices with the same CDC are likely to offer the same resistance to current flow for corresponding terminals.

Keywords: bipartite (canonical) double covering, main eigenspace, comain graphs, walk matrix, two-fold isomorphism, SSP model.

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