{"title":"Spectral extrema of graphs with fixed size: Forbidden triangles and pentagons","authors":"Shuchao Li , Yuantian Yu","doi":"10.1016/j.disc.2024.114151","DOIUrl":null,"url":null,"abstract":"<div><p>Nosal (1970) and Nikiforov (2002) showed that if graph <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-free of size <em>m</em>, then the spectral radius of <em>G</em> satisfies <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>m</mi></mrow></msqrt></math></span>, equality holds if and only if <em>G</em> is a complete bipartite graph. Lin, Ning and Wu (2021) extended this result as: If <em>G</em> is a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-free non-bipartite graph of size <em>m</em>, then <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msqrt></math></span>, equality holds if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>. This result was extended by Li, Peng (2022) and Sun, Li (2023), independently, as the following: If <em>G</em> is a <span><math><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>}</mo></math></span>-free non-bipartite graph with <em>m</em> edges, then <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>)</mo><mo>)</mo></math></span>, equality holds if and only if <em>m</em> is odd and <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>)</mo></math></span> is obtained from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></math></span> by replacing one of its edges by a path of length 4. This upper bound could be attained only if <em>m</em> is odd, since the extremal graph <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>)</mo></math></span> is well-defined only in this case. Thus, it is interesting to determine the spectral extremal graph when <em>m</em> is even. Sun and Li (2023) proposed the following question: Determine the graphs attaining the maximum spectral radius over all <span><math><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>}</mo></math></span>-free non-bipartite graphs of even size <em>m</em>. In this contribution, we answer this question for <span><math><mi>m</mi><mo>≥</mo><mn>150</mn></math></span>. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of Ning and Zhai's triangle counting lemma in terms of both eigenvalues and the size of a graph, together with the eigenvector method from Lou, Lu and Huang (2023).</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-10","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/S0012365X24002826","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Nosal (1970) and Nikiforov (2002) showed that if graph G is -free of size m, then the spectral radius of G satisfies , equality holds if and only if G is a complete bipartite graph. Lin, Ning and Wu (2021) extended this result as: If G is a -free non-bipartite graph of size m, then , equality holds if and only if . This result was extended by Li, Peng (2022) and Sun, Li (2023), independently, as the following: If G is a -free non-bipartite graph with m edges, then , equality holds if and only if m is odd and , where is obtained from by replacing one of its edges by a path of length 4. This upper bound could be attained only if m is odd, since the extremal graph is well-defined only in this case. Thus, it is interesting to determine the spectral extremal graph when m is even. Sun and Li (2023) proposed the following question: Determine the graphs attaining the maximum spectral radius over all -free non-bipartite graphs of even size m. In this contribution, we answer this question for . Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of Ning and Zhai's triangle counting lemma in terms of both eigenvalues and the size of a graph, together with the eigenvector method from Lou, Lu and Huang (2023).
Nosal (1970) 和 Nikiforov (2002) 发现,如果图 G 是大小为 m 的无 C3 图,则 G 的谱半径满足 λ(G)≤m,且只有当且仅当 G 是一个完整的双向图时,等式才成立。Lin、Ning 和 Wu (2021) 将这一结果扩展为:如果 G 是大小为 m 的无 C3 非双向图,那么只有当 G≅C5 时,λ(G)≤m-1,等式成立。李鹏(2022)和孙莉(2023)分别将这一结果扩展如下:如果 G 是一个有 m 条边的无{C3,C5}非双面图,那么λ(G)≤λ(S3(K2,m-32)),当且仅当 m 为奇数且 G≅S3(K2,m-32),其中 S3(K2,m-32) 是通过将 K2,m-32 的一条边替换为长度为 4 的路径得到的。只有当 m 为奇数时才能达到这个上限,因为极值图 S3(K2,m-32) 只有在这种情况下才定义明确。因此,确定 m 为偶数时的谱极值图是很有意义的。孙和李(2023 年)提出了以下问题:在本论文中,我们将回答 m≥150 时的这个问题。我们的证明技术主要基于图的特征值的 Cauchy 交错定理,并借助于宁和翟在特征值和图的大小方面的三角形计数法,以及 Lou, Lu 和 Huang (2023) 的特征向量法。
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.