The least signless Laplacian eigenvalue of $\{C_3,C_5\}$-free graphs

Qi Zhou

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arXiv · 2610.00875

The least signless Laplacian eigenvalue of $\{C_3,C_5\}$-free graphs

Qi Zhou

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Abstract

Brandt [Discrete Math. 183 (1998) 17--25] conjectured that the least signless Laplacian eigenvalue of every regular triangle-free graph of order $n$ is at most $4n/25$. Using flag algebras, Balogh, Clemen, Lidický, Norin and Volec [SIAM J. Discrete Math. 37 (2023) 1173--1179] established the stronger bound $15n/94$ for all triangle-free graphs. We investigate the effect of additionally excluding pentagons and prove that every $\{C_3,C_5\}$-free graph $G$ of order $n$ satisfies $\qmin(G)<0.0569n$, without any regularity assumption. The proof combines seven-vertex flag inequalities with local Rayleigh constraints that retain the least eigenvalue throughout the counting argument. An exact integer certificate establishes the required inequality.

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Qi Zhou. 2026-10-01. The least signless Laplacian eigenvalue of $\{C_3,C_5\}$-free graphs. https://arxiv.org/abs/2610.00875

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