{"title":"改进的树宽光谱下界","authors":"Tatsuya Gima , Tesshu Hanaka , Kohei Noro , Hirotaka Ono , Yota Otachi","doi":"10.1016/j.ipl.2024.106536","DOIUrl":null,"url":null,"abstract":"<div><div>We show that for every <em>n</em>-vertex graph with at least one edge, its treewidth is greater than or equal to <span><math><mi>n</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mo>(</mo><mi>Δ</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where Δ and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [<em>Inf. Process. Lett.</em>, 2003] and the subsequent one by the authors of the present paper [<em>IEICE Trans. Inf. Syst.</em>, 2024]. The new lower bound is <em>almost</em> tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An improved spectral lower bound of treewidth\",\"authors\":\"Tatsuya Gima , Tesshu Hanaka , Kohei Noro , Hirotaka Ono , Yota Otachi\",\"doi\":\"10.1016/j.ipl.2024.106536\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We show that for every <em>n</em>-vertex graph with at least one edge, its treewidth is greater than or equal to <span><math><mi>n</mi><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mo>(</mo><mi>Δ</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where Δ and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [<em>Inf. Process. Lett.</em>, 2003] and the subsequent one by the authors of the present paper [<em>IEICE Trans. Inf. Syst.</em>, 2024]. The new lower bound is <em>almost</em> tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.</div></div>\",\"PeriodicalId\":56290,\"journal\":{\"name\":\"Information Processing Letters\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-10-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Information Processing Letters\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0020019024000668\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INFORMATION SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information Processing Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020019024000668","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
We show that for every n-vertex graph with at least one edge, its treewidth is greater than or equal to , where Δ and are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only 1 less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues.
期刊介绍：
Information Processing Letters invites submission of original research articles that focus on fundamental aspects of information processing and computing. This naturally includes work in the broadly understood field of theoretical computer science; although papers in all areas of scientific inquiry will be given consideration, provided that they describe research contributions credibly motivated by applications to computing and involve rigorous methodology. High quality experimental papers that address topics of sufficiently broad interest may also be considered.
Since its inception in 1971, Information Processing Letters has served as a forum for timely dissemination of short, concise and focused research contributions. Continuing with this tradition, and to expedite the reviewing process, manuscripts are generally limited in length to nine pages when they appear in print.
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