Fowler, P.W. orcid.org/0000-0003-2106-1104, Gauci, J.B., Goedgebeur, J. et al. (2 more authors) (2020) Existence of regular nut graphs for degree at most 11. Discussiones Mathematicae Graph Theory, 40 (2). pp. 533-557. ISSN 1234-3099
Abstract
A nut graph is a singular graph with one-dimensional kernel and corresponding eigenvector with no zero elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here we solve the problem for all remaining cases d ≤ 11 and determine the complete lists of all d-regular nut graphs of order n for small values of d and n. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any d-regular nut graph of order n, another d-regular nut graph of order n+2d. If we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all d-regular nut graphs of higher orders is established. For even d the orders n are indeed consecutive, while for odd d the orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 The Author(s) and the Faculty of Mathematics, Computer Science and Econometrics - University of Zielona Góra. Uploaded with the publisher's permission. For re-use permissions, please contact the publisher. |
Keywords: | nut graph; core graph; regular graph; nullity |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > Department of Chemistry (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 02 Mar 2020 10:57 |
Last Modified: | 06 Mar 2020 07:38 |
Published Version: | https://www.dmgt.uz.zgora.pl/publish/volume.php?ID... |
Status: | Published |
Publisher: | University of Zielona Góra |
Refereed: | Yes |
Identification Number: | 10.7151/dmgt.2283 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:157831 |