ISSN 2590-9770 
The Art of Discrete and Applied Mathematics 5 (2022) #P2.01 
https://doi.org/10.26493/2590-9770.1403.1b1 
(Also available at http://adam-journal.eu) 
On 12-regular nut graphs* 
Nino Bašic´† FAMNIT&IAM, Universityof Primorska, 6000Koper, Slovenia,and Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia 
Martin Knor ‡ 
Departmentof Mathematics,Facultyof Civil Engineering, Slovak Universityof Technology in Bratislava, Radlinského 11, 810 05, Bratislava, Slovakia 
Riste Škrekovski § Faculty of Information Studies, 8000 Novo mesto, Slovenia,and FMF, University of Ljubljana, 1000 Ljubljana, Slovenia, and FAMNIT, Universityof Primorska, 6000Koper, Slovenia 
Received8February 2021, accepted29May 2021, published online11May 2022 
Abstract 
Anut graph is a simple graph whose adjacencymatrix is singular with1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020,Fowleret al. characterised for each d .{3, 4,..., 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show that there exists a 12-regular nut graph of order n if and only if n . 16.We also present a resultbywhich there are infinitely manycirculant nut graphsofdegree d . 0 (mod 4) and no circulant nut graphs of degree d . 2 (mod 4). The former result partially resolves a questionbyFowler et al. onexistenceofvertex-transitive nut graphsof order n and degree 
d.We conclude the paper with problems, conjectures and ideas for further work. 
Keywords: Nutgraph, adjacency matrix, singular matrix, coregraph,Fowler construction,regular graph. Math. Subj. Class.: 05C50, 15A18 
*We would like to thank the two anonymous referees for their comments which helped to improve the presen­tation of the paper. 
†The work of the first author is supported in part by the Slovenian Research Agency(research program P1­0294 and research projects J1-9187, J1-1691, N1-0140 and J1-2481). 
‡The second author acknowledges partial supportby Slovak research grants APVV-15-0220, APVV-17-0428, VEGA 1/0206/20 and VEGA 1/0238/19. §The researchofthe third authorwas partially supportedbytheSlovenian ResearchAgency(ARRS), research 
program P1-0383 and research projects J1-1692 and J1-8130. E-mail addresses: nino.basic@famnit.upr.si (Nino Baši ´
c), knor@math.sk (Martin Knor), skrekovski@gmail.com (Riste Škrekovski) 
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 

ISSN 2590-9770 
The Art of Discrete and Applied Mathematics 5 (2022) #P2.01 
https://doi.org/10.26493/2590-9770.1403.1b1 
(Dostopno tudi na http://adam-journal.eu) 

O12-regularnih orešnih grafih* 
Nino Bašic´† FAMNIT&IAM, Universityof Primorska, 6000Koper, Slovenia,and Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia 
Martin Knor ‡ 
Departmentof Mathematics,Facultyof Civil Engineering, Slovak Universityof Technology in Bratislava, Radlinského 11, 810 05, Bratislava, Slovakia 
Riste Škrekovski § Faculty of Information Studies, 8000 Novo mesto, Slovenia,and FMF, University of Ljubljana, 1000 Ljubljana, Slovenia, and FAMNIT, Universityof Primorska, 6000Koper, Slovenia 
Prejeto 8. februarja 2021, sprejeto 29. maja 2021, objavljeno na spletu 11. maja 2022 
Povzetek 
Orešni graf je enostaven graf, katerega matrika sosednosti je singularna z 1-dimenzio­nalnim jedrom, tako da ustrezni lastni vektor nima niˇ
celnih vnosov. Leta 2020 soFowler idr. karakterizirali za vsak d .{3, 4,..., 11} vse vrednosti n, tako da obstaja d-regularni orešni graf reda n. V priˇcujoˇcem prispevku razrešimo prvi odprt primer d = 12, tj. pokažemo, da obstaja 12-regularen orešni graf reda n ˇce je n . 16. Pred­
ce in samo ˇstavimo tudi rezultat, da obstaja neskonˇ
cno mnogo cirkulantov stopnje d . 0 (mod 4), ki so hkrati orešni grafi, ter noben cirkulant stopnje d . 2 (mod 4), ki bi bil orešni graf. Prvi rezultat delno odgovori na vprašanjeFowlerja idr. o obstoju vozlišˇ
cno tranzitivnih orešnih grafov reda n in stopnje d. Prispevek zakljuˇ
cimo s problemi, domnevami in idejami za nadaljnje delo. 
Kljuˇcne besede:Orešnigraf, matrika sosednosti, singularna matrika, srediˇcnigraf,Fowlerjevakon­strukcija, regularen graf. Math. Subj. Class.: 05C50, 15A18 
*Zahvaljujemo se anonimnima recenzentoma zakomentarje,ki so pripomoglikboljši predstavitvi ˇclanka. 
†Prviavtorje delno podprts straniJavne agencijeza raziskovalnodejavnost RepublikeSlovenije (raziskovalni program P1-0294 ter raziskovalni projekti J1-9187, J1-1691, N1-0140 in J1-2481). 
‡Drugiavtorjedelno podportsslovaškimi raziskovalnimi dotacijami APVV-15-0220, APVV-17-0428, VEGA 1/0206/20 in VEGA 1/0238/19. §Raziskavalno delo tretjega avtorja je delno podprla Javna agencija za raziskovalno dejavnost Republike 
Slovenije (ARRS), raziskovalni program P1-0383 ter raziskovalna projekta J1-1692 in J1-8130. E-poštni naslovi: nino.basic@famnit.upr.si (Nino Baši ´
c), knor@math.sk (Martin Knor), skrekovski@gmail.com (Riste Škrekovski) 
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/