The Spectrum of Triangle-free Graphs

Understand

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 $\left\lfloor\frac{n}{2}\right\rfloor$ spanning at most $n^2/50$ edges.