The Spectrum of Triangle-free Graphs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866914328547950592 |
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| author | Balogh, József Clemen, Felix Christian Lidický, Bernard Norin, Sergey Volec, Jan |
| author_facet | Balogh, József Clemen, Felix Christian Lidický, Bernard Norin, Sergey Volec, Jan |
| contents | 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 Erdős:
(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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_00093 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Spectrum of Triangle-free Graphs Balogh, József Clemen, Felix Christian Lidický, Bernard Norin, Sergey Volec, Jan Combinatorics 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 Erdős: (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. |
| title | The Spectrum of Triangle-free Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2204.00093 |