A sharp bound for the least signless Laplacian eigenvalue of triangle-free graphs

Abstract

We prove that every finite simple triangle-free graph $G$ on $n\geq1$ vertices satisfies $q_{\min}(G)\leq7n/50$, where $q_{\min}(G)$ is the least eigenvalue of its signless Laplacian matrix $Q_G=D_G+A_G$, with $D_G$ the diagonal matrix of vertex degrees and $A_G$ the adjacency matrix. The constant is sharp, since the Higman--Sims graph and its balanced blow-ups attain equality, as shown by de Lima, Nikiforov and Oliveira (2016). This proves the bound conjectured by Balogh, Clemen, Lidick\'y, Norin and Volec (2023). The proof uses five test column vectors associated with an ordered edge to obtain a matrix Rayleigh inequality. Our use of local Rayleigh inequalities together with flag-algebra semidefinite programming is inspired by Zhou (2026). We use semidefinite programming and rational approximation to find the required positive semidefinite matrices, and complete the proof by exact correction and verification.