{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T14:31:26Z","timestamp":1787668286205,"version":"build-2736575974"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T00:00:00Z","timestamp":1785456000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T00:00:00Z","timestamp":1785456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s12190-026-02868-y","type":"journal-article","created":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T13:09:16Z","timestamp":1785503356000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-backtracking cycle estimates in supersingular isogeny graphs"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-1298-0587","authenticated-orcid":false,"given":"Mohammed","family":"El Baraka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Siham","family":"Ezzouak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,31]]},"reference":[{"key":"2868_CR1","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/BF02126799","volume":"8","author":"A. Lubotzky","year":"1988","unstructured":"Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8, 261\u2013277 (1988). https:\/\/doi.org\/10.1007\/BF02126799","journal-title":"Combinatorica"},{"issue":"1","key":"2868_CR2","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1006\/jctb.1994.1054","volume":"62","author":"M. Morgenstern","year":"1994","unstructured":"Morgenstern, M.: Existence and explicit constructions of $$q+1$$ regular Ramanujan graphs for every prime power $$q$$. J. Comb. Theory Ser. B 62(1), 44\u201362 (1994). https:\/\/doi.org\/10.1006\/jctb.1994.1054","journal-title":"J. Comb. Theory, Ser. B"},{"issue":"1","key":"2868_CR3","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1090\/S0273-0979-1990-15918-X","volume":"23","author":"A.K. Pizer","year":"1990","unstructured":"Pizer, A.K.: Ramanujan graphs and Hecke operators. Bull. Am. Math. Soc. 23(1), 127\u2013137 (1990). https:\/\/doi.org\/10.1090\/S0273-0979-1990-15918-X","journal-title":"Bull. Am. Math. Soc."},{"key":"2868_CR4","doi-asserted-by":"publisher","unstructured":"Vign\u00e9ras, M.-F.: Arithm\u00e9tique des alg\u00e8bres de quaternions. In: Lecture Notes in Mathematics, vol. 800. Springer, Berlin (1980). https:\/\/doi.org\/10.1007\/BFb0091027","DOI":"10.1007\/BFb0091027"},{"issue":"1","key":"2868_CR5","doi-asserted-by":"publisher","first-page":"25","DOI":"10.30538\/psrp-odam2025.0108","volume":"8","author":"S. Ali","year":"2025","unstructured":"Ali, S., Jamil, M.K.: A novel resolvability parameter: face metric dimension and its applications. Open J. Discrete Appl. Math. 8(1), 25\u201334 (2025). https:\/\/doi.org\/10.30538\/psrp-odam2025.0108","journal-title":"Open J. Discrete Appl. Math."},{"issue":"11","key":"2868_CR6","doi-asserted-by":"publisher","first-page":"103009","DOI":"10.1016\/j.asej.2024.103009","volume":"15","author":"P. Liu","year":"2024","unstructured":"Liu, P., Asim, M.H., Ali, S., Azeem, M., Almohsen, B.: Utilizing the lexicographic max product of the picture fuzzy graph in human trafficking. Ain. Shams Eng. J. 15(11), 103009 (2024). https:\/\/doi.org\/10.1016\/j.asej.2024.103009","journal-title":"Ain. Shams Eng. J."},{"issue":"4","key":"2868_CR7","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1142\/S0219199707002551","volume":"9","author":"N. Alon","year":"2007","unstructured":"Alon, N., Benjamini, I., Lubetzky, E., Sodin, S.: Non-backtracking random walks mix faster. Commun. Contemp. Math. 9(4), 585\u2013603 (2007). https:\/\/doi.org\/10.1142\/S0219199707002551","journal-title":"Commun. Contemp. Math."},{"issue":"6","key":"2868_CR8","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1142\/S0129167X92000357","volume":"3","author":"H. Bass","year":"1992","unstructured":"Bass, H.: The Ihara\u2013Selberg zeta function of a tree lattice. Int. J. Math. 3(6), 717\u2013797 (1992). https:\/\/doi.org\/10.1142\/S0129167X92000357","journal-title":"Int. J. Math."},{"issue":"4","key":"2868_CR9","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1142\/S0129167X90000204","volume":"1","author":"K.-I. Hashimoto","year":"1990","unstructured":"Hashimoto, K.-I.: On zeta and $$L$$-functions of finite graphs. Int. J. Math. 1(4), 381\u2013396 (1990). https:\/\/doi.org\/10.1142\/S0129167X90000204","journal-title":"Int. J. Math."},{"issue":"3","key":"2868_CR10","doi-asserted-by":"publisher","first-page":"219","DOI":"10.2969\/jmsj\/01830219","volume":"18","author":"Y. Ihara","year":"1966","unstructured":"Ihara, Y.: On discrete subgroups of the two by two projective linear group over $$p$$-adic fields. J. Math. Soc. Jpn. 18(3), 219\u2013235 (1966). https:\/\/doi.org\/10.2969\/jmsj\/01830219","journal-title":"J. Math. Soc. Jpn."},{"issue":"1","key":"2868_CR11","first-page":"7","volume":"7","author":"M. Kotani","year":"2000","unstructured":"Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7\u201325 (2000)","journal-title":"J. Math. Sci., Univ. Tokyo"},{"issue":"1","key":"2868_CR12","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1006\/aima.1996.0050","volume":"121","author":"H.M. Stark","year":"1996","unstructured":"Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121(1), 124\u2013165 (1996). https:\/\/doi.org\/10.1006\/aima.1996.0050","journal-title":"Adv. Math."},{"key":"2868_CR13","doi-asserted-by":"publisher","unstructured":"Terras, A.: Zeta functions of graphs: a stroll through the garden. In: Cambridge Studies in Advanced Mathematics, vol. 128. Cambridge University Press (2010). https:\/\/doi.org\/10.1017\/CBO9780511760426","DOI":"10.1017\/CBO9780511760426"},{"key":"2868_CR14","doi-asserted-by":"publisher","unstructured":"Delfs, C., Galbraith, S.D.: Computing isogenies between supersingular elliptic curves over $$\\mathbb{F}_p$$, designs, codes and cryptography. 78, 425\u2013440 (2016). https:\/\/doi.org\/10.1007\/s10623-014-0010-1","DOI":"10.1007\/s10623-014-0010-1"},{"key":"2868_CR15","unstructured":"Kohel, D.R.: Endomorphism Rings of Elliptic Curves Over Finite Fields, Ph.D. thesis, University of California, Berkeley, 1996. https:\/\/math.berkeley.edu\/publications\/endomorphism-rings-elliptic-curves-over-finite-fields,"},{"key":"2868_CR16","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s00145-007-9002-x","volume":"22","author":"D.X. Charles","year":"2009","unstructured":"Charles, D.X., Goren, E.Z., Lauter, K.E.: Cryptographic hash functions from expander graphs. J. Cryptol. 22, 93\u2013113 (2009). https:\/\/doi.org\/10.1007\/s00145-007-9002-x","journal-title":"J. Cryptol."},{"key":"2868_CR17","doi-asserted-by":"publisher","unstructured":"De Feo, L.: Mathematics of isogeny based cryptography. arXiv preprint arXiv:1711.04062. 2017. https:\/\/doi.org\/10.48550\/arXiv.1711.04062","DOI":"10.48550\/arXiv.1711.04062"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02868-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02868-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02868-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T13:57:25Z","timestamp":1787666245000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02868-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,31]]},"references-count":17,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["2868"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02868-y","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,31]]},"assertion":[{"value":"10 May 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"213"}}