BALOGH, J., et al. THE SPECTRUM OF TRIANGLE-FREE GRAPHS. SIAM Journal on Discrete Mathematics. 2023, 37(2), 1173-1179. ISSN 0895-4801. DOI 10.1137/22M150767X.
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.
