  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 Erd\H{o}s:
  (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.
