Abstract
Denote by the smallest eigenvalue of the signless Laplacian matrix of an -vertex graph . Brandt conjectured in 1997 that for regular triangle-free graphs . We prove a stronger result: If is a triangle-free graph, then . Brandt’s conjecture is a subproblem of two famous conjectures of Erdős: (1) Sparse-half-conjecture: Every -vertex triangle-free graph has a subset of vertices of size spanning at most edges. (2) Every -vertex triangle-free graph can be made bipartite by removing at most edges. In our proof we use linear algebraic methods to upper bound 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.
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