A note on Cayley nut graphs whose degree is divisible by four

Damnjanović, Ivan

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Title:	A note on Cayley nut graphs whose degree is divisible by four
Authors:	ID Damnjanović, Ivan (Author)
Files:	ADAM_Damnjanovic_2026.pdf (404,59 KB) MD5: BAFD582FC6DA4E44610C47B9D76DEAFA
Language:	English
Work type:	Article
Typology:	1.01 - Original Scientific Article
Organization:	ZUP - University of Primorska Press
Abstract:	A nut graph is a nontrivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. Fowler et al. in 2020 proved that there is a d-regular vertex-transitive nut graph of order n only if 4 ∣ d, 2 ∣ n, n ≥ d + 4 or d≡₄2, 4 ∣ n and n ≥ d + 6. It was recently shown that there exists a d-regular circulant nut graph of order n if and only if 4 ∣ d, 2 ∣ n, d > 0, together with n ≥ d + 4 if d≡₈4 and n ≥ d + 6 if 8 ∣ d, as well as (n, d) ≠ (16, 8) (in the paper from 2024). In this paper, we demonstrate the existence of a d-regular Cayley nut graph of order n for each n and d with 4 ∣ d, d > 0 and 2 ∣ n, n ≥ d + 4, thereby finding all the orders attainable by a Cayley nut graph, or vertex-transitive nut graph, with a fixed degree divisible by four.
Keywords:	nut graph, Cayley graph, vertex-transitive graph, circulant graph, graph spectrum, graph eigenvalue
Publication status:	Published
Publication version:	Version of Record
Publication date:	03.02.2026
Publisher:	Založba Univerze na Primorskem
Year of publishing:	2026
Number of pages:	6 str.
Numbering:	Vol. 9, no. 2, [article no.] P2.02
PID:	20.500.12556/RUP-22838
UDC:	51
eISSN:	2590-9770
DOI:	10.26493/2590-9770.1662.4e9
Publication date in RUP:	23.03.2026
Views:	210
Downloads:	4
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Record is a part of a journal

Title:	The Art of Discrete and Applied Mathematics
Publisher:	Založba Univerze na Primorskem
ISSN:	2590-9770

Document is financed by a project

Funder:	Ministry of Science, Technological Development and Innovation of the Republic of Serbia
Project number:	451-03-137/2025-03/200102

Funder:	Science Fund of the Republic of Serbia
Project number:	#6767

Licences

License:	CC BY 4.0, Creative Commons Attribution 4.0 International

Link:	http://creativecommons.org/licenses/by/4.0/
Description:	This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

Language:	Slovenian
Title:	Notica o Cayleyjevih orešnih grafih, katerih stopnja je deljiva s štiri
Abstract:	Orešni graf je netrivialen enostaven graf, katerega matrika sosednosti ima enodimen-zionalen ničelni prostor, razpet s polnim vektorjem. Fowler in sod. so leta 2020 dokazali, da obstaja d-regularen točkovno tranzitiven orešni graf redanle, če je 4\|d, 2\|n, n≥d + 4 ali d ≡ 42, 4\|n in n≥d + 6. Nedavno je bilo pokazano, da obstaja d-regularen cirkulanten orešni graf reda n natanko tedaj, ko je 4\|d, 2\|n, d >0, skupaj z dodatnimi pogoji n≥d + 4, ce d ≡ 84, in n≥d + 6, če 8\|d, ter (n, d)̸ = (16,8) (v članku iz leta 2024). V tem članku pokažemo, da za vsak n in d, za katera velja 4\|d, d >0 in 2\|n, n≥d + 4, obstaja d-regularen Cayleyjev orešni graf redan, s čimer določimo vse možne rede, ki jih lahko doseže Cayleyjev ali točkovno tranzitiven orešni graf s fiksno stopnjo, deljivo s štiri.
Keywords:	orešni graf, Cayleyjev graf, točkovno tranzitiven graf, cirkulantni graf, spekter grafa, lastne vrednosti grafa

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