{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T21:09:34Z","timestamp":1776114574664,"version":"3.50.1"},"reference-count":40,"publisher":"Elsevier BV","issue":"8","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/100007224","name":"National Foundation for Science &amp; Technology Development","doi-asserted-by":"publisher","award":["101.04-2024.07"],"award-info":[{"award-number":["101.04-2024.07"]}],"id":[{"id":"10.13039\/100007224","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Discrete Mathematics"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1016\/j.disc.2026.115109","type":"journal-article","created":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T19:28:31Z","timestamp":1773775711000},"page":"115109","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Log-concavity of the independence polynomials of W graphs"],"prefix":"10.1016","volume":"349","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9875-6516","authenticated-orcid":false,"given":"Do Trong","family":"Hoang","sequence":"first","affiliation":[]},{"given":"Vadim E.","family":"Levit","sequence":"additional","affiliation":[]},{"given":"Eugen","family":"Mandrescu","sequence":"additional","affiliation":[]},{"given":"My Hanh","family":"Pham","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.disc.2026.115109_br0010","first-page":"15","article-title":"The vertex independence sequence of a graph is not constrained","volume":"58","author":"Alavi","year":"1987","journal-title":"Congr. Numer."},{"key":"10.1016\/j.disc.2026.115109_br0020","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1002\/jgt.22724","article-title":"Independent set and matching permutations","volume":"99","author":"Ball","year":"2022","journal-title":"J. Graph Theory"},{"key":"10.1016\/j.disc.2026.115109_br0030","first-page":"31","article-title":"Some common properties for regularizable graphs, edge-critical graphs and B-graphs","volume":"12","author":"Berge","year":"1982","journal-title":"Ann. Discrete Math."},{"key":"10.1016\/j.disc.2026.115109_br0040","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1023\/B:JACO.0000030703.39946.70","article-title":"On the location of roots of independence polynomials","volume":"19","author":"Brown","year":"2004","journal-title":"J. Algebr. Comb."},{"key":"10.1016\/j.disc.2026.115109_br0050","first-page":"215","article-title":"On well-covered 3 polytopes","volume":"25","author":"Campbell","year":"1988","journal-title":"Ars Comb."},{"key":"10.1016\/j.disc.2026.115109_br0060","first-page":"509","article-title":"Unimodality of very well-covered graphs","volume":"97A","author":"Chen","year":"2010","journal-title":"Ars Comb."},{"key":"10.1016\/j.disc.2026.115109_br0070","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.jctb.2006.06.001","article-title":"The roots of the independence polynomial of a claw-free graph","volume":"97","author":"Chudnovsky","year":"2007","journal-title":"J. Comb. Theory, Ser. B"},{"key":"10.1016\/j.disc.2026.115109_br0080","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/0012-365X(82)90215-1","article-title":"Very well-covered graphs","volume":"42","author":"Favaron","year":"1982","journal-title":"Discrete Math."},{"key":"10.1016\/j.disc.2026.115109_br0090","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1006\/jctb.1993.1005","article-title":"A characterization of well-covered graphs of girth 5 or greater","volume":"57","author":"Finbow","year":"1993","journal-title":"J. Comb. Theory, Ser. B"},{"key":"10.1016\/j.disc.2026.115109_br0100","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1007\/BF01844162","article-title":"On the corona of two graphs","volume":"4","author":"Frucht","year":"1970","journal-title":"Aequ. Math."},{"key":"10.1016\/j.disc.2026.115109_br0110","first-page":"97","article-title":"Generalizations of the matching polynomial","volume":"24","author":"Gutman","year":"1983","journal-title":"Util. Math."},{"key":"10.1016\/j.disc.2026.115109_br0120","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1007\/BF01877590","article-title":"Theory of monomer-dimer systems","volume":"25","author":"Heilmann","year":"1972","journal-title":"Commun. Math. Phys."},{"key":"10.1016\/j.disc.2026.115109_br0140","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s00013-022-01724-2","article-title":"The size of Betti tables of edge ideal of clique corona graphs","volume":"118","author":"Hoang","year":"2022","journal-title":"Arch. Math."},{"key":"10.1016\/j.disc.2026.115109_br0150","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10801-015-0631-0","article-title":"A characterization of triangle-free Gorenstein graphs and Cohen\u2013Macaulayness of second powers of edge ideals","volume":"43","author":"Hoang","year":"2016","journal-title":"J. Algebr. Comb."},{"key":"10.1016\/j.disc.2026.115109_br0130","author":"Hoang"},{"key":"10.1016\/j.disc.2026.115109_br0160","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0012-365X(94)90163-5","article-title":"Clique polynomials and independent set polynomials of graphs","volume":"125","author":"Hoede","year":"1994","journal-title":"Discrete Math."},{"key":"10.1016\/j.disc.2026.115109_br0170","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1090\/S0894-0347-2012-00731-0","article-title":"Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs","volume":"25","author":"Huh","year":"2012","journal-title":"J. Am. Math. Soc."},{"key":"10.1016\/j.disc.2026.115109_br0180","doi-asserted-by":"crossref","DOI":"10.26493\/1855-3974.3207.2ad","article-title":"The independence polynomial of trees is not always log-concave starting from order 26","author":"Kadrawi","year":"2025","journal-title":"Ars Math. Contemp."},{"key":"10.1016\/j.disc.2026.115109_br0190","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1080\/01621459.1971.10482273","article-title":"Some results for discrete unimodality","volume":"334","author":"Keilson","year":"1971","journal-title":"J. Am. Stat. Assoc."},{"key":"10.1016\/j.disc.2026.115109_br0230","first-page":"195","article-title":"A family of well-covered graphs with unimodal independence polynomials","volume":"165","author":"Levit","year":"2003","journal-title":"Congr. Numer."},{"key":"10.1016\/j.disc.2026.115109_br0250","first-page":"73","article-title":"Very well-covered graphs with log-concave independence polynomials","volume":"20","author":"Levit","year":"2004","journal-title":"Carpath. J. Math."},{"key":"10.1016\/j.disc.2026.115109_br0240","series-title":"Proceedings of the 1st International Conference on Algebraic Informatics","first-page":"233","article-title":"The independence polynomial of a graph - a survey","author":"Levit","year":"2005"},{"key":"10.1016\/j.disc.2026.115109_br0200","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.ejc.2005.04.007","article-title":"Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture","volume":"27","author":"Levit","year":"2006","journal-title":"Eur. J. Comb."},{"key":"10.1016\/j.disc.2026.115109_br0210","series-title":"Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge","first-page":"243","article-title":"Independence polynomials and the unimodality conjecture for very well-covered, quasi-regularizable, and perfect graphs","author":"Levit","year":"2007"},{"key":"10.1016\/j.disc.2026.115109_br0220","first-page":"108","article-title":"A family of graphs whose independence polynomials are both palindromic and unimodal","volume":"23","author":"Levit","year":"2007","journal-title":"Carpath. J. Math."},{"key":"10.1016\/j.disc.2026.115109_br0260","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1007\/s00373-017-1834-z","article-title":"The Roller-Coaster conjecture revisited","volume":"33","author":"Levit","year":"2017","journal-title":"Graphs Comb."},{"key":"10.1016\/j.disc.2026.115109_br0270","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.ejc.2018.02.021","article-title":"1-well-covered graphs revisited","volume":"80","author":"Levit","year":"2019","journal-title":"Eur. J. Comb."},{"key":"10.1016\/j.disc.2026.115109_br0280","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s00373-002-0515-7","article-title":"Independence sequences of well-covered graphs: non-unimodality and the Roller-Coaster conjecture","volume":"19","author":"Michael","year":"2003","journal-title":"Graphs Comb."},{"key":"10.1016\/j.disc.2026.115109_br0310","series-title":"W2 graphs and strongly well-covered graphs: two well-covered graph subclasses","author":"Pinter","year":"1991"},{"key":"10.1016\/j.disc.2026.115109_br0300","first-page":"159","article-title":"Planar regular one-well-covered graphs","volume":"91","author":"Pinter","year":"1992","journal-title":"Congr. Numer."},{"key":"10.1016\/j.disc.2026.115109_br0290","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1002\/jgt.3190190108","article-title":"A class of planar well-covered graphs with girth four","volume":"19","author":"Pinter","year":"1995","journal-title":"J. Graph Theory"},{"key":"10.1016\/j.disc.2026.115109_br0320","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0021-9800(70)80011-4","article-title":"Some covering concepts in graphs","volume":"8","author":"Plummer","year":"1970","journal-title":"J. Comb. Theory"},{"key":"10.1016\/j.disc.2026.115109_br0330","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/16073606.1993.9631737","article-title":"Well-covered graphs: survey","volume":"16","author":"Plummer","year":"1993","journal-title":"Quaest. Math."},{"key":"10.1016\/j.disc.2026.115109_br0340","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1111\/j.1749-6632.1989.tb16434.x","article-title":"Log-concave and unimodal sequences in algebra, combinatorics, and geometry","volume":"576","author":"Stanley","year":"1989","journal-title":"Ann. N.Y. Acad. Sci."},{"key":"10.1016\/j.disc.2026.115109_br0350","series-title":"On some subclasses of well-covered graphs","author":"Staples","year":"1975"},{"key":"10.1016\/j.disc.2026.115109_br0360","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1002\/jgt.3190030211","article-title":"On some subclasses of well-covered graphs","volume":"3","author":"Staples","year":"1979","journal-title":"J. Graph Theory"},{"key":"10.1016\/j.disc.2026.115109_br0370","first-page":"199","article-title":"On the well\u2013coveredness of products of graphs","volume":"33","author":"Topp","year":"1992","journal-title":"Ars Comb."},{"key":"10.1016\/j.disc.2026.115109_br0380","article-title":"Monomial Algebras","volume":"vol. 238","author":"Villarreal","year":"2001"},{"key":"10.1016\/j.disc.2026.115109_br0390","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.dam.2016.01.030","article-title":"Clique cover products and unimodality of independence polynomials","volume":"206","author":"Zhu","year":"2016","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/j.disc.2026.115109_br0400","first-page":"35","article-title":"Log-concavity of independence polynomials of some kinds of trees","volume":"342","author":"Zhu","year":"2019","journal-title":"Appl. Math. Comput."}],"container-title":["Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0012365X26001330?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0012365X26001330?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T19:56:41Z","timestamp":1776110201000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0012365X26001330"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":40,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["S0012365X26001330"],"URL":"https:\/\/doi.org\/10.1016\/j.disc.2026.115109","relation":{},"ISSN":["0012-365X"],"issn-type":[{"value":"0012-365X","type":"print"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Log-concavity of the independence polynomials of Wp graphs","name":"articletitle","label":"Article Title"},{"value":"Discrete Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.disc.2026.115109","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"115109"}}