J. Balogh, F. Clemen, Bernard Lidick'y, S. Norin, Jan Volec
{"title":"无三角图的谱","authors":"J. Balogh, F. Clemen, Bernard Lidick'y, S. Norin, Jan Volec","doi":"10.1137/22m150767x","DOIUrl":null,"url":null,"abstract":"Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \\leq \\frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \\leq \\frac{15n}{94}<\\frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erd\\H{o}s: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\\lceil\\frac{n}{2}\\rceil$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"44 1","pages":"1173-1179"},"PeriodicalIF":0.0000,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"The Spectrum of Triangle-Free Graphs\",\"authors\":\"J. Balogh, F. Clemen, Bernard Lidick'y, S. Norin, Jan Volec\",\"doi\":\"10.1137/22m150767x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \\\\leq \\\\frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \\\\leq \\\\frac{15n}{94}<\\\\frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erd\\\\H{o}s: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\\\\lceil\\\\frac{n}{2}\\\\rceil$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":\"44 1\",\"pages\":\"1173-1179\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-03-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/22m150767x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/22m150767x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \leq \frac{15n}{94}<\frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erd\H{o}s: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\lceil\frac{n}{2}\rceil$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.