https://m2.mtmt.hu/ hun 2026-06-25 07:31 JournalArticle 36396168 VALIDATED true Funding Agency and Grant Number: Yongtao Li was supported by the Postdoctoral Fellowship Program of CPSF (No. GZC20233196), Lihua Feng was supported by the National Natural Science Foundation of China (Nos. 12271527 and 12471022), and Yuejian Peng was supported by the National Natural Sci [GZC20233196]; Postdoctoral Fellowship Program of CPSF [11931002, 12371327]; National Natural Science Foundation of China [2025JJ30003]; National Natural Science Foundation of Hunan Province Funding text: The authors would like to thank Xiaocong He and Loujun Yu for carefully reading an early manuscript of this paper. The authors also show their great gratitude to anonymous referees for valuable suggestions, which considerably improve the presentation of the paper. Yongtao Li was supported by the Postdoctoral Fellowship Program of CPSF (No. GZC20233196), Lihua Feng was supported by the National Natural Science Foundation of China (Nos. 12271527 and 12471022), and Yuejian Peng was supported by the National Natural Science Foundation of Hunan Province (No. 2025JJ30003) and the National Natural Science Foundation of China (Nos. 11931002 and 12371327). 0 false 2025-12-31T10:45:49.853+0000 2025-12-19T00:04:58.234+0000 2025-10-22T16:21:55.422+0000 true 10066925 Nemec Évi /api/admin/10066925 Admin Nemec Évi (MTMT Központi 3, admin) true 2025-12-31T10:44:17.166+0000 2025-12-19T00:04:57.369+0000 true 521 Szuper Admin /api/admin/521 Admin Szuper Admin (admin) true false true true 24 24 /api/publicationtype/24 PublicationType Folyóiratcikk 1 true 24 JournalArticle true 10000059 Article true 24 24 /api/publicationtype/24 PublicationType Folyóiratcikk 1 true 24 JournalArticle /api/subtype/10000059 Szakcikk SubType Szakcikk (Folyóiratcikk) 101 true 10000059 true 1 /api/category/1 Category Tudományos true 1 Li, Yongtao true true /api/journal/2050556 REVIEWED JOURNAL OF GRAPH THEORY 0364-9024 1097-0118 true false 2050556 false 2050556 true 0364-9024 1097-0118 Journal FOREIGN 18 2025 A well-known theorem of Mantel states that every n $n$-vertex graph with more than & LeftFloor; n 2 / 4 & RightFloor; $\lfloor {n}<^>{2}\unicode{x02215}4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erd & odblac;s, Faudree, and Rousseau (1992) showed that a graph on n $n$ vertices with more than & LeftFloor; n 2 / 4 & RightFloor; $\lfloor {n}<^>{2}\unicode{x02215}4\rfloor $ edges contains at least 2 & LeftFloor; n / 2 & RightFloor; + 1 $2\lfloor n\unicode{x02215}2\rfloor +1$ edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erd & odblac;s, Faudree, and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every n $n$-vertex graph G $G$ with lambda ( G ) >= & LeftFloor; n 2 / 4 & RightFloor; $\lambda (G)\ge \sqrt{\lfloor {n}<^>{2}\unicode{x02215}4\rfloor }$ contains at least 2 & LeftFloor; n / 2 & RightFloor; - 1 $2\lfloor n\unicode{x02215}2\rfloor -1$ triangular edges, unless G $G$ is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation-stability can be used to revisit a conjecture of Erd & odblac;s concerning the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khad & zcaron;iivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order n $n$ of the spectral extremal graph when we forbid the friendship graph as a substructure. We drop the condition that requires the order n $n$ to be sufficiently large, which was investigated by Cioab & abreve; et al. (2020) using the triangle removal lemma. Thirdly, this method can be utilized to deduce the classical stability for odd cycles, and it gives more concise bounds on parameters. Finally, supersaturation stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023). 2025 true true true false false false NONE 2025-12-31 false 0 0 0 0 0 0 0 0 0 0 0 0 7 7 Q1 false false false false 2026-01-20T17:23:43.979+0000 4 20 0 3 0 true Language 10002 /api/language/10002 Angol Angol English true 2 true PersonAuthorship 128264981 /api/authorship/128264981 Li, Yongtao 1 0.33333334 true false false Li Yongtao true false false AuthorshipType 1 /api/authorshiptype/1 Szerző 0 true 0 true false true PersonAuthorship 128264982 /api/authorship/128264982 Feng, Lihua ✉ 2 0.33333334 false false true Feng Lihua true false false AuthorshipType 1 /api/authorshiptype/1 Szerző 0 true 0 true false true PersonAuthorship 128264983 /api/authorship/128264983 Peng, Yuejian ✉ 3 0.33333334 false true true Peng Yuejian true false false AuthorshipType 1 /api/authorshiptype/1 Szerző 0 true 0 true false true PublicationIdentifier 30050046 /api/publicationidentifier/30050046 DOI: 10.1002/jgt.23280 PlainSource 6 /api/publicationsource/6 DOI PublicationSourceType 10001 /api/publicationsourcetype/10001 DOI true true true DOI DOI https://doi.org/@@@ true true 6 true IDENTICAL 10.1002/jgt.23280 https://doi.org/10.1002/jgt.23280 false true PublicationIdentifier 30050047 /api/publicationidentifier/30050047 WoS: 001529706900001 PlainSource 1 /api/publicationsource/1 WoS PublicationSourceType 10003 /api/publicationsourcetype/10003 Indexelő adatbázis false true true WoS WoS https://www.webofscience.com/wos/woscc/full-record/@@@ true true 1 true IDENTICAL 001529706900001 https://www.webofscience.com/wos/woscc/full-record/001529706900001 false true PublicationIdentifier 30050050 /api/publicationidentifier/30050050 Scopus: 105010747542 PlainSource 3 /api/publicationsource/3 Scopus PublicationSourceType 10003 /api/publicationsourcetype/10003 Indexelő adatbázis false true true Scopus Scopus http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-@@@ true true 3 true IDENTICAL 105010747542 http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-105010747542 false true Keyword 2611 /api/keyword/2611 DENSITY true 2611 true Keyword 3236 /api/keyword/3236 STABILITY true 3236 true Keyword 9505 /api/keyword/9505 BOUNDS true 9505 true Keyword 9510 /api/keyword/9510 GRAPHS true 9510 true Keyword 1078635 /api/keyword/1078635 EDGES true 1078635 true Keyword 1141067 /api/keyword/1141067 extremal graph theory true 1141067 true Keyword 1167344 /api/keyword/1167344 CLIQUES true 1167344 true Keyword 1193269 /api/keyword/1193269 Spectral radius true 1193269 true Keyword 1215547 /api/keyword/1215547 TRIANGLES true 1215547 true Keyword 2576025 /api/keyword/2576025 MINIMUM NUMBER true true SjrRating 11573484 /api/sjrrating/11573484 sjr:D1 (2025) Scopus - Discrete Mathematics and Combinatorics JOURNAL OF GRAPH THEORY 0364-9024 1097-0118 5 0.1 journal RatingType 10002 /api/ratingtype/10002 sjr sjr true true ClassificationExternal 2607 /api/classificationexternal/2607 Scopus - Discrete Mathematics and Combinatorics true 2607 true D1 FROM_LAST_YEAR true true Reference 69231422 /api/reference/69231422 1. 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(2025) JOURNAL OF GRAPH THEORY 0364-9024 1097-0118<div class="JournalArticle Publication long-list"> <div class="authors"> <img title="Idézőközlemény" style="float: left" src="/frontend/resources/grid/publication-citation-icon.png"> <div class="autype autype0"> <span class="author-name" >Li Yongtao </span> ;&nbsp;&nbsp;&nbsp; <span class="author-name" >Feng Lihua ✉ </span> ;&nbsp;&nbsp;&nbsp; <span class="author-name" >Peng Yuejian ✉ </span> </div> </div> <div class="title"><a href="/gui2/?mode=browse&params=publication;36396168" target="_blank">A Spectral Erdős-Faudree-Rousseau Theorem</a></div> <div> <span class="journal-title">JOURNAL OF GRAPH THEORY</span> <span class="journal-issn">(<a target="_blank" href="https://portal.issn.org/resource/ISSN/0364-9024">0364-9024</a> <a target="_blank" href="https://portal.issn.org/resource/ISSN/1097-0118">1097-0118</a>)</span>: <span class="journal-volume"></span> <span class="page"> 18 p. </span> <span class="year">(2025)</span> </div> <div class="pub-footer"> <span class="language" xmlns="http://www.w3.org/1999/html">Nyelv: Angol | </span> <span class="identifiers"> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="10.1002/jgt.23280" target="_blank" href="https://doi.org/10.1002/jgt.23280"> DOI </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="001529706900001" target="_blank" href="https://www.webofscience.com/wos/woscc/full-record/001529706900001"> WoS </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="105010747542" target="_blank" href="http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-105010747542"> Scopus </a> </span> </span> <div class="publication-citation"> <a target="_blank" href="/api/publication?cond=citations.related;eq;36396168&sort=publishedYear,desc&sort=title"> Idézett közlemények száma: 7 </a> </div> <div class="mtid"><span class="long-pub-mtid">Közlemény: 36396168</span> | <span class="status-data status-VALIDATED"> Egyeztetett </span> Idéző | <span class="type-subtype">Folyóiratcikk ( Szakcikk ) </span> | <span class="pub-category">Tudományos</span> | <span class="publication-sourceOfData">WOS TAGGED</span> </div> <div class="lastModified">Utolsó módosítás: 2025.12.19. 01:04 Szuper Admin (admin) </div> <pre class="comment" style="margin-top: 0; margin-bottom: 0;"><u>Megjegyzés</u>: Funding Agency and Grant Number: Yongtao Li was supported by the Postdoctoral Fellowship Program of CPSF (No. GZC20233196), Lihua Feng was supported by the National Natural Science Foundation of China (Nos. 12271527 and 12471022), and Yuejian Peng was supported by the National Natural Sci [GZC20233196]; Postdoctoral Fellowship Program of CPSF [11931002, 12371327]; National Natural Science Found...</pre> </div></div>