A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue

• DOI: 10.48550/arxiv.2502.05520
• Publication Date: 2025-02-08
• Authors: Jack H. Koolen; Hao Ge; Chenhui Lv; Qianqian Yang
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 05C50, 05C75, 05E30

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ large minimum degree. this study, we depart from the conventional use of theory instead employ novel approach that combines Bose-Laskar type argument to derive insights into $\mu$-bounded fixed eigenvalue. Our method establishes reasonable bound Note local distance-regular are $\mu$-bounded. We apply these results characterize structure for any graph classical parameters $(D,b,\alpha,\beta)$. Consequently, show parameter $\alpha$ is bounded cubic polynomial in $b$ if $D \geq 9$ $b 2$. Also $\alpha \leq 2$ =2$ 12$.