{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T09:28:19Z","timestamp":1771406899073,"version":"3.50.1"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline1.png\"\/>\n                        <jats:tex-math>$-2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline2.png\"\/>\n                        <jats:tex-math>$(\\! -\\lambda ^*, -2)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline3.png\"\/>\n                        <jats:tex-math>$\\lambda ^* = ho ^{1\/2} + ho ^{-1\/2} \\approx 2.01980$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline4.png\"\/>\n                        <jats:tex-math>$ho$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is the unique real root of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline5.png\"\/>\n                        <jats:tex-math>$x^3 = x + 1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline6.png\"\/>\n                        <jats:tex-math>$(\\! -\\lambda , -2)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for any constant\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100278_inline7.png\"\/>\n                        <jats:tex-math>$\\lambda \\gt 2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1017\/s0963548325100278","type":"journal-article","created":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T11:50:13Z","timestamp":1764330613000},"page":"207-229","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5134-1911","authenticated-orcid":false,"given":"Hricha","family":"Acharya","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03efmqc40","id-type":"ROR","asserted-by":"publisher"}],"name":"Arizona State University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2946-7347","authenticated-orcid":false,"given":"Zilin","family":"Jiang","sequence":"additional","affiliation":[{"name":"Arizona State University"}]}],"member":"56","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"S0963548325100278_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-023-00002-1"},{"key":"S0963548325100278_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(76)90162-9"},{"key":"S0963548325100278_ref4","first-page":"147","article-title":"Some results on generalized line graphs","volume":"2","author":"Cvetkovi\u0107","year":"1980","journal-title":"C. R. Math. Rep. Acad. Sci. Canada"},{"key":"S0963548325100278_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(77)90027-1"},{"key":"S0963548325100278_ref2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1992-1134718-6"},{"key":"S0963548325100278_ref13","article-title":"On the smallest eigenvalues of \n\n\n\n$3$\n\n\n-colorable graphs","author":"Jiang","year":"2025","journal-title":"Proc. Amer. Math. Soc."},{"key":"S0963548325100278_ref7","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1996.0026"},{"key":"S0963548325100278_ref5","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190050408"},{"key":"S0963548325100278_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2023.113745"},{"key":"S0963548325100278_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0092290"},{"key":"S0963548325100278_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01904834"},{"key":"S0963548325100278_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(82)90023-4"},{"key":"S0963548325100278_ref8","unstructured":"[8] Hoffman, A. J. (1970) \u22121 \u2212 $\\sqrt{2}$ ? In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969). Gordon and Breach, pp. 173\u2013176."},{"key":"S0963548325100278_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067367"},{"key":"S0963548325100278_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511751752"},{"key":"S0963548325100278_ref17","unstructured":"[17] Seidel, J. J. (1973) On two-graphs and Shult\u2019s characterization of symplectic and orthogonal geometries over GF (2). T. H.-Report, No. 73-WSK-02, Department of Mathematics, Technological University Eindhoven, Eindhoven."},{"key":"S0963548325100278_ref11","doi-asserted-by":"publisher","DOI":"10.1017\/fms.2025.10110"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100278","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T08:53:54Z","timestamp":1771404834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100278\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,28]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0963548325100278"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100278","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,28]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}