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 Erdős:
(1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\left\lfloor\frac{n}{2}\right\rfloor$ 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.
From: Jan Volec [view email]
[v1]
Thu, 31 Mar 2022 21:09:49 UTC (79 KB)
[v2]
Fri, 22 Apr 2022 14:22:41 UTC (81 KB)
[v3]
Wed, 4 Jan 2023 18:23:41 UTC (84 KB)
[v4]
Wed, 9 Sep 2026 01:40:20 UTC (83 KB)
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