THE SPECTRUM OF TRIANGLE-FREE GRAPHS

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F23%3A00368031" target="_blank" >RIV/68407700:21340/23:00368031 - isvavai.cz</a>
Result on the web
<a href="http://hdl.handle.net/10467/112906" target="_blank" >http://hdl.handle.net/10467/112906</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/22M150767X" target="_blank" >10.1137/22M150767X</a>

Result language
angličtina
Original language name
THE SPECTRUM OF TRIANGLE-FREE GRAPHS
Original language description
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)<= 4n/25. We prove a stronger result: If G is a triangle-free graph, then q_n(G) <= 15n/94 < 4n/25. Brandt's conjecture is a subproblem of two famous conjectures of Erdos: (1) Sparse-half-conjecture: Every n-vertex triangle-free graph has a subset of vertices of size the ceiling of n/2 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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GM23-06815M" target="_blank" >GM23-06815M: Extremal and probabilistic combinatorics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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