{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T05:49:40Z","timestamp":1782107380115,"version":"3.54.5"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T00:00:00Z","timestamp":1782086400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T00:00:00Z","timestamp":1782086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100015029","name":"Vicerrector\u00eda de Investigaci\u00f3n y Estudios de Posgrado, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla","doi-asserted-by":"publisher","award":["00695-PV\/2026"],"award-info":[{"award-number":["00695-PV\/2026"]}],"id":[{"id":"10.13039\/501100015029","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2026,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$|z+1\/3|=1\/3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\/<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\/<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin,\n                    <jats:italic>Trees with non log-concave independent set sequences<\/jats:italic>\n                    , arXiv:2502.10654v1, 2025].\n                  <\/jats:p>","DOI":"10.1007\/s00373-026-03054-4","type":"journal-article","created":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T05:42:25Z","timestamp":1782106945000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linear Recurrences for Non-Log-Concave Independence Polynomials of Trees"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5069-1415","authenticated-orcid":false,"given":"C\u00e9sar","family":"Bautista-Ramos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carlos","family":"Guill\u00e9n-Galv\u00e1n","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paulino","family":"G\u00f3mez-Salgado","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,22]]},"reference":[{"key":"3054_CR1","first-page":"15","volume":"58","author":"Y Alavi","year":"1987","unstructured":"Alavi, Y., Malde, P.J., Schwenk, A.J., Erd\u00f6s, P.: The vertex independence sequence of a graph is not constrained. Congr. Numer. 58, 15\u201323 (1987)","journal-title":"Congr. Numer."},{"key":"3054_CR2","unstructured":"Arocha, J.L.: Propiedades del polinomio independiente de un grafo. Cienc. Mat. (Havana) V, 103\u2013110 (1984)"},{"key":"3054_CR3","unstructured":"Bautista-Ramos, C.: Multiple breaks of log-concavity in the independence polynomials of trees. arXiv:2511.00334 (2025)"},{"key":"3054_CR4","first-page":"213","volume":"1","author":"S Beraha","year":"1978","unstructured":"Beraha, S., Kahane, J., Weiss, N.: Limits of zeros of recursively defined families of polynomials. Stud. Found. Comb. 1, 213\u2013232 (1978)","journal-title":"Stud. Found. Comb."},{"key":"3054_CR5","unstructured":"Brassard, G., Bratley, P.: Fundamentals of Algorithmics. Prentice-Hall, Inc. (1996)"},{"key":"3054_CR6","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1006\/jabr.1995.1189","volume":"175","author":"U Cerruti","year":"1995","unstructured":"Cerruti, U., Vaccarino, F.: $$R$$-algebras of linear recurrent sequences. J. Algebra 175, 332\u2013338 (1995)","journal-title":"J. Algebra"},{"key":"3054_CR7","unstructured":"Galvin, D.: Trees with non log-concave independent set sequences. arXiv:2511.00334 (2025)"},{"issue":"1","key":"3054_CR8","first-page":"97","volume":"24","author":"I Gutman","year":"1983","unstructured":"Gutman, I., Harary, F.: Generalizations of the matching polynomial. Utilitas Math. 24(1), 97\u2013106 (1983)","journal-title":"Utilitas Math."},{"issue":"4","key":"3054_CR9","doi-asserted-by":"publisher","first-page":"03","DOI":"10.26493\/1855-3974.3207.2ad","volume":"25","author":"O Kadrawi","year":"2025","unstructured":"Kadrawi, O., Levit, V.E.: The independence polynomial of trees is not always log-concave starting from order 26. Ars Math. Contemp. 25(4), 03 (2025)","journal-title":"Ars Math. Contemp."},{"key":"3054_CR10","doi-asserted-by":"crossref","unstructured":"Kadrawi, O., Levit, V.E., Yosef, R., Mizrachi, M.: On computing of independence polynomials of trees. In: \u00d6zger, F. (ed.) Recent Research in Polynomials. IntechOpen (2023)","DOI":"10.5772\/intechopen.1001130"},{"key":"3054_CR11","unstructured":"Knuth, D.E.: The Art of Computer Programming: Fundamental Algorithms, vol. 1. Addison-Wesley (1997)"},{"key":"3054_CR12","doi-asserted-by":"crossref","unstructured":"Levit, V.E., Mandrescu, E.: Very well-covered graphs with log-concave independence polynomials. Carpathian J. Math. 20(1), 73\u201380 (2004)","DOI":"10.1016\/j.ejc.2018.02.021"},{"key":"3054_CR13","unstructured":"Ramos, E., Sun, S.: An AI enhanced approach to the tree unimodality conjecture. arXiv:2510.18826 (2025)"}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-026-03054-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-026-03054-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-026-03054-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T05:42:36Z","timestamp":1782106956000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-026-03054-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,22]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["3054"],"URL":"https:\/\/doi.org\/10.1007\/s00373-026-03054-4","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"value":"0911-0119","type":"print"},{"value":"1435-5914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,22]]},"assertion":[{"value":"5 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 June 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 June 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"59"}}