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The walks and CDC of graphs with the same main eigenspace
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Sciriha, Irene and Collins, Luke (2023) The walks and CDC of graphs with the same main eigenspace. Discussiones Mathematicae Graph Theory, 43 (2). pp. 507-532. doi:10.7151/dmgt.2386 ISSN 1234-3099.
An open access version can be found in:
Official URL: http://dx.doi.org/10.7151/dmgt.2386
Abstract
The main eigenvalues of a graph
are those eigenvalues of the
-adjacency matrix
with a corresponding eigenspace not orthogonal to
. The principal main eigenvector associated with a main eigenvalue is the orthogonal projection of the corresponding eigenspace onto
. 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
is the direct product
. 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
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Discussiones Mathematicae Graph Theory | ||||||||
Publisher: | University of Zielona Góra | ||||||||
ISSN: | 1234-3099 | ||||||||
Official Date: | 2023 | ||||||||
Dates: |
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Volume: | 43 | ||||||||
Number: | 2 | ||||||||
Page Range: | pp. 507-532 | ||||||||
DOI: | 10.7151/dmgt.2386 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Open Access Version: |
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