{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T02:04:35Z","timestamp":1767924275204,"version":"3.49.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004329","name":"Slovenian Research Agency","doi-asserted-by":"crossref","award":["P1-0294"],"award-info":[{"award-number":["P1-0294"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100016047","name":"Science Fund of the Republic of Serbia","doi-asserted-by":"crossref","award":["#6767"],"award-info":[{"award-number":["#6767"]}],"id":[{"id":"10.13039\/501100016047","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-circulant graph is a graph that admits a cyclic group of automorphisms having <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> vertex orbits of equal size. It is not difficult to observe that there exists no cubic 1-circulant nut graph or cubic 2-circulant nut graph, while the full classification of all the cubic 3-circulant nut graphs was recently obtained (Damnjanovi\u0107 et al. in Electron J Comb 31(2):P2.31, 2024). Here, we investigate the existence of cubic <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-circulant nut graphs for <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell \\ge 4$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and show that there is no cubic 4-circulant nut graph or cubic 5-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-circulant nut graphs for any fixed <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell \\in \\{6, 7 \\}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mn>6<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>7<\/mml:mn>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> or <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ell \\ge 9$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>9<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s40314-025-03218-7","type":"journal-article","created":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T20:36:37Z","timestamp":1746563797000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On cubic polycirculant nut graphs"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6555-8668","authenticated-orcid":false,"given":"Nino","family":"Ba\u0161i\u0107","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7329-1759","authenticated-orcid":false,"given":"Ivan","family":"Damnjanovi\u0107","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,6]]},"reference":[{"key":"3218_CR1","unstructured":"Ba\u0161i\u0107 N, Damnjanovi\u0107 I On cubic polycirculant nut graphs: supplementary material (GitHub repository). https:\/\/github.com\/nbasic\/cubic-polycirculant-nuts"},{"key":"3218_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/s10801-025-01389-4","author":"N Ba\u0161i\u0107","year":"2025","unstructured":"Ba\u0161i\u0107 N, Fowler PW (2025) Nut graphs with a given automorphism group. J Algebr Comb. https:\/\/doi.org\/10.1007\/s10801-025-01389-4","journal-title":"J Algebr Comb"},{"key":"3218_CR3","doi-asserted-by":"publisher","unstructured":"Ba\u0161i\u0107 N, Knor M, \u0160krekovski R (2022) On $$12$$-regular nut graphs. Art Discrete Appl Math. https:\/\/doi.org\/10.26493\/2590-9770.1403.1b1","DOI":"10.26493\/2590-9770.1403.1b1"},{"key":"3218_CR4","doi-asserted-by":"publisher","unstructured":"Ba\u0161i\u0107 N, Fowler PW, Pisanski T (2024) Vertex and edge orbits in nut graphs. Electron J Comb 31(2):P2.38. https:\/\/doi.org\/10.37236\/12619","DOI":"10.37236\/12619"},{"key":"3218_CR5","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.tcs.2012.01.018","volume":"502","author":"G Brinkmann","year":"2013","unstructured":"Brinkmann G, Van Cleemput N, Pisanski T (2013) Generation of various classes of trivalent graphs. Theor Comput Sci 502:16\u201329. https:\/\/doi.org\/10.1016\/j.tcs.2012.01.018","journal-title":"Theor Comput Sci"},{"key":"3218_CR6","unstructured":"Coolsaet K, Fowler PW, Goedgebeur J (2018) Generation and properties of nut graphs. MATCH Commun Math Comput Chem 80:423\u2013444. https:\/\/match.pmf.kg.ac.rs\/electronic_versions\/Match80\/n2\/match80n2_423-444.pdf"},{"key":"3218_CR7","volume-title":"Spectra of graphs: theory and applications","author":"D Cvetkovi\u0107","year":"1995","unstructured":"Cvetkovi\u0107 D, Doob M, Sachs H (1995) Spectra of graphs: theory and applications. Barth Verlag, Leipzig, J. A"},{"key":"3218_CR8","unstructured":"Damnjanovi\u0107 I (2023) A note on Cayley nut graphs whose degree is divisible by four. arXiv:2305.18658"},{"issue":"24","key":"3218_CR9","doi-asserted-by":"publisher","first-page":"8331","DOI":"10.2298\/FIL2324331D","volume":"37","author":"I Damnjanovi\u0107","year":"2023","unstructured":"Damnjanovi\u0107 I (2023) Two families of circulant nut graphs. Filomat 37(24):8331\u20138360. https:\/\/doi.org\/10.2298\/FIL2324331D","journal-title":"Filomat"},{"key":"3218_CR10","doi-asserted-by":"publisher","unstructured":"Damnjanovi\u0107 I (2024) Complete resolution of the circulant nut graph order-degree existence problem. Ars Math Contemp. https:\/\/doi.org\/10.26493\/1855-3974.3009.6df","DOI":"10.26493\/1855-3974.3009.6df"},{"key":"3218_CR11","unstructured":"Damnjanovi\u0107 I (2024) On the null spaces of quartic circulant graphs. Discrete Math Chem (in press)"},{"key":"3218_CR12","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.laa.2021.10.006","volume":"633","author":"I Damnjanovi\u0107","year":"2022","unstructured":"Damnjanovi\u0107 I, Stevanovi\u0107 D (2022) On circulant nut graphs. Linear Algebra Appl 633:127\u2013151. https:\/\/doi.org\/10.1016\/j.laa.2021.10.006","journal-title":"Linear Algebra Appl"},{"key":"3218_CR13","doi-asserted-by":"publisher","unstructured":"Damnjanovi\u0107 I, Ba\u0161i\u0107 N, Pisanski T, \u017ditnik A (2024) Classification of cubic tricirculant nut graphs. Electron J Comb. 31(2):P2.31 https:\/\/doi.org\/10.37236\/12668","DOI":"10.37236\/12668"},{"issue":"5","key":"3218_CR14","doi-asserted-by":"publisher","DOI":"10.1063\/1.4863559","volume":"140","author":"PW Fowler","year":"2014","unstructured":"Fowler PW, Pickup BT, Todorova TZ, Borg M, Sciriha I (2014) Omni-conducting and omni-insulating molecules. J Chem Phys 140(5):054115. https:\/\/doi.org\/10.1063\/1.4863559","journal-title":"J Chem Phys"},{"key":"3218_CR15","doi-asserted-by":"publisher","first-page":"533","DOI":"10.7151\/dmgt.2283","volume":"40","author":"PW Fowler","year":"2020","unstructured":"Fowler PW, Gauci JB, Goedgebeur J, Pisanski T, Sciriha I (2020) Existence of regular nut graphs for degree at most $$11$$. Discuss Math Gr Theory 40:533\u2013557. https:\/\/doi.org\/10.7151\/dmgt.2283","journal-title":"Discuss Math Gr Theory"},{"key":"3218_CR16","unstructured":"Fowler PW, Pisanski T, Ba\u0161i\u0107 N (2021) Charting the space of chemical nut graphs. MATCH Commun Math Comput Chem 86(3):519\u2013538 https:\/\/match.pmf.kg.ac.rs\/electronic_versions\/Match86\/n3\/match86n3_519-538.pdf"},{"issue":"2","key":"3218_CR17","doi-asserted-by":"publisher","first-page":"321","DOI":"10.2298\/AADM190517028G","volume":"17","author":"JB Gauci","year":"2023","unstructured":"Gauci JB, Pisanski T, Sciriha I (2023) Existence of regular nut graphs and the Fowler construction. Appl Anal Discrete Math 17(2):321\u2013333. https:\/\/doi.org\/10.2298\/AADM190517028G","journal-title":"Appl Anal Discrete Math"},{"issue":"3","key":"3218_CR18","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1561\/0100000006","volume":"2","author":"RM Gray","year":"2006","unstructured":"Gray RM (2006) Toeplitz and circulant matrices: a review. Found Trends Commun Inf Theory 2(3):155\u2013239. https:\/\/doi.org\/10.1561\/0100000006","journal-title":"Found Trends Commun Inf Theory"},{"key":"3218_CR19","unstructured":"Jameson GJO The cyclotomic polynomials. https:\/\/www.maths.lancs.ac.uk\/~jameson\/cyp.pdf"},{"key":"3218_CR20","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1006\/eujc.2000.0390","volume":"21","author":"A Malni\u010d","year":"2000","unstructured":"Malni\u010d A, Nedela R, \u0160koviera M (2000) Lifting graph automorphisms by voltage assignments. Eur J Comb 21:927\u2013947. https:\/\/doi.org\/10.1006\/eujc.2000.0390","journal-title":"Eur J Comb"},{"key":"3218_CR21","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1023\/B:JACO.0000047294.42633.25","volume":"20","author":"A Malni\u010d","year":"2004","unstructured":"Malni\u010d A, Maru\u0161i\u010d D, Poto\u010dnik P (2004) Elementary Abelian covers of graphs. J Algebraic Comb 20:71\u201397. https:\/\/doi.org\/10.1023\/B:JACO.0000047294.42633.25","journal-title":"J Algebraic Comb"},{"key":"3218_CR22","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.jsc.2013.09.003","volume":"60","author":"BD McKay","year":"2014","unstructured":"McKay BD, Piperno A (2014) Practical graph isomorphism II. J Symb Comput 60:94\u2013112. https:\/\/doi.org\/10.1016\/j.jsc.2013.09.003","journal-title":"J Symb Comput"},{"issue":"3\u20135","key":"3218_CR23","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1016\/j.disc.2005.09.053","volume":"307","author":"T Pisanski","year":"2007","unstructured":"Pisanski T (2007) A classification of cubic bicirculants. Discrete Math 307(3\u20135):567\u2013578. https:\/\/doi.org\/10.1016\/j.disc.2005.09.053","journal-title":"Discrete Math"},{"key":"3218_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8364-1","volume-title":"Configurations from a Graphical Viewpoint","author":"T Pisanski","year":"2013","unstructured":"Pisanski T, Servatius B (2013) Configurations from a Graphical Viewpoint. Springer, New York, Birkh\u00e4user Advanced Texts Basler Lehrb\u00fccher. https:\/\/doi.org\/10.1007\/978-0-8176-8364-1"},{"key":"3218_CR25","doi-asserted-by":"publisher","unstructured":"Poto\u010dnik P, Toledo M (2020) Classification of cubic vertex-transitive tricirculants. Ars Math Contemp 18(1):1\u201331. https:\/\/doi.org\/10.26493\/1855-3974.1815.b52","DOI":"10.26493\/1855-3974.1815.b52"},{"key":"3218_CR26","first-page":"97","volume":"52","author":"I Sciriha","year":"1997","unstructured":"Sciriha I (1997) On the coefficient of $$\\lambda $$ in the characteristic polynomial of singular graphs. Util Math 52:97\u2013111","journal-title":"Util Math"},{"key":"3218_CR27","unstructured":"Sciriha I (1998a) On singular line graphs of trees. Congr Numer 135:73\u201391"},{"key":"3218_CR28","doi-asserted-by":"publisher","unstructured":"Sciriha I (1998b) On the construction of graphs of nullity one. Discrete Math 181(1\u20133):193\u2013211. https:\/\/doi.org\/10.1016\/S0012-365X(97)00036-8","DOI":"10.1016\/S0012-365X(97)00036-8"},{"issue":"5","key":"3218_CR29","first-page":"167","volume":"20","author":"I Sciriha","year":"1999","unstructured":"Sciriha I (1999) The two classes of singular line graphs of trees. Rend Semin Mat Messina Ser II 20(5):167\u2013180","journal-title":"Rend Semin Mat Messina Ser II"},{"key":"3218_CR30","doi-asserted-by":"crossref","unstructured":"Sciriha I (2007) A characterization of singular graphs. Electron J Linear Algebra 16:451\u2013462. https:\/\/eudml.org\/doc\/129125","DOI":"10.13001\/1081-3810.1215"},{"key":"3218_CR31","doi-asserted-by":"publisher","unstructured":"Sciriha I (2008) Coalesced and embedded nut graphs in singular graphs. Ars Math Contemp 1:20\u201331. https:\/\/doi.org\/10.26493\/1855-3974.20.7cc","DOI":"10.26493\/1855-3974.20.7cc"},{"issue":"5","key":"3218_CR32","doi-asserted-by":"publisher","first-page":"1763","DOI":"10.1021\/ci700097j","volume":"47","author":"I Sciriha","year":"2007","unstructured":"Sciriha I, Fowler PW (2007) Nonbonding orbitals in fullerenes: nuts and cores in singular polyhedral graphs. J Chem Inf Model 47(5):1763\u20131775. https:\/\/doi.org\/10.1021\/ci700097j","journal-title":"J Chem Inf Model"},{"issue":"2\u20133","key":"3218_CR33","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/j.disc.2006.11.040","volume":"308","author":"I Sciriha","year":"2008","unstructured":"Sciriha I, Fowler PW (2008) On nut and core singular fullerenes. Discrete Math 308(2\u20133):267\u2013276. https:\/\/doi.org\/10.1016\/j.disc.2006.11.040","journal-title":"Discrete Math"},{"key":"3218_CR34","unstructured":"Sciriha I, Farrugia A (2021) From nut graphs to molecular structure and conductivity. Mathematical chemistry monographs, vol. 23. University of Kragujevac, Kragujevac"},{"key":"3218_CR35","first-page":"257","volume":"54","author":"I Sciriha","year":"1998","unstructured":"Sciriha I, Gutman I (1998) Nut graphs: maximally extending cores. Util Math 54:257\u2013272","journal-title":"Util Math"},{"key":"3218_CR36","unstructured":"The Sage Developers: SageMath, the Sage Mathematics Software System (Version 9.5) (2022). https:\/\/www.sagemath.org"},{"key":"3218_CR37","unstructured":"Van\u00a0Cleemput N (2014a) Sequence A243391. In: The on-line encyclopedia of integer sequences. The OEIS Foundation Inc. https:\/\/oeis.org\/A243391"},{"key":"3218_CR38","unstructured":"Van\u00a0Cleemput N (2014b) Sequence A243393. In: The on-line encyclopedia of integer sequences. The OEIS Foundation Inc. https:\/\/oeis.org\/A243393"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03218-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03218-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03218-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T14:11:18Z","timestamp":1747663878000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03218-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,6]]},"references-count":38,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["3218"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03218-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,6]]},"assertion":[{"value":"25 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"265"}}