{"title":"非二部图上的分解族和谱极值问题","authors":"Longfei Fang , Michael Tait , Mingqing Zhai","doi":"10.1016/j.disc.2025.114527","DOIUrl":null,"url":null,"abstract":"<div><div>Given a graph family <span><math><mi>H</mi></math></span> with <span><math><msub><mrow><mi>min</mi></mrow><mrow><mi>H</mi><mo>∈</mo><mi>H</mi></mrow></msub><mo></mo><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></math></span>. Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all <em>n</em>-vertex <span><math><mi>H</mi></math></span>-free graphs, respectively. Denote by <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>) the set of extremal graphs with respect to <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>). A fundamental problem in extremal spectral graph theory asks which graph <em>H</em> satisfies <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>.</div><div>Wang et al. (2023) <span><span>[43]</span></span> proved that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite graph <em>H</em> with <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. In this paper, we use decomposition family defined by Simonovits to give a characterization of which graph families <span><math><mi>H</mi></math></span> satisfy <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo><</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>⌋</mo></math></span>. Using this result, we show that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite family <span><math><mi>H</mi></math></span> with <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo><</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>⌋</mo></math></span>.</div><div>As an application, we completely determine <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> for <em>n</em> sufficiently large, where <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> denotes a finite graph family which consists of <em>k</em> edge-disjoint <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-chromatic color-critical graphs <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. This result strengthens a theorem of Győri, who settled the case that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. Furthermore, we determine <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> for <em>n</em> sufficiently large.</div><div>Finally, two related problems are proposed for further research.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114527"},"PeriodicalIF":0.7000,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Decomposition family and spectral extremal problems on non-bipartite graphs\",\"authors\":\"Longfei Fang , Michael Tait , Mingqing Zhai\",\"doi\":\"10.1016/j.disc.2025.114527\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Given a graph family <span><math><mi>H</mi></math></span> with <span><math><msub><mrow><mi>min</mi></mrow><mrow><mi>H</mi><mo>∈</mo><mi>H</mi></mrow></msub><mo></mo><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></math></span>. Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all <em>n</em>-vertex <span><math><mi>H</mi></math></span>-free graphs, respectively. Denote by <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>) the set of extremal graphs with respect to <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> (resp. <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>). A fundamental problem in extremal spectral graph theory asks which graph <em>H</em> satisfies <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>.</div><div>Wang et al. (2023) <span><span>[43]</span></span> proved that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite graph <em>H</em> with <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. In this paper, we use decomposition family defined by Simonovits to give a characterization of which graph families <span><math><mi>H</mi></math></span> satisfy <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo><</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>⌋</mo></math></span>. Using this result, we show that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> for <em>n</em> sufficiently large and any finite family <span><math><mi>H</mi></math></span> with <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo><</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac><mo>⌋</mo></math></span>.</div><div>As an application, we completely determine <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> for <em>n</em> sufficiently large, where <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> denotes a finite graph family which consists of <em>k</em> edge-disjoint <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-chromatic color-critical graphs <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. This result strengthens a theorem of Győri, who settled the case that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. Furthermore, we determine <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> for <em>n</em> sufficiently large.</div><div>Finally, two related problems are proposed for further research.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"348 10\",\"pages\":\"Article 114527\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-04-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X25001359\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X25001359","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Decomposition family and spectral extremal problems on non-bipartite graphs
Given a graph family with . Let and be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all n-vertex -free graphs, respectively. Denote by (resp. ) the set of extremal graphs with respect to (resp. ). A fundamental problem in extremal spectral graph theory asks which graph H satisfies .
Wang et al. (2023) [43] proved that for n sufficiently large and any finite graph H with . In this paper, we use decomposition family defined by Simonovits to give a characterization of which graph families satisfy . Using this result, we show that for n sufficiently large and any finite family with .
As an application, we completely determine for n sufficiently large, where denotes a finite graph family which consists of k edge-disjoint -chromatic color-critical graphs . This result strengthens a theorem of Győri, who settled the case that . Furthermore, we determine for n sufficiently large.
Finally, two related problems are proposed for further research.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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