A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue
Jack H. Koolen, Hong-Jun Ge, Chenhui Lv, Qianqian Yang
TL;DR
The paper develops a Bose–Laskar–type approach fused with Hoffman graph theory to study μ-bounded graphs with fixed smallest eigenvalue, deriving structural constraints and a bound on the minimum degree. By constructing associated Hoffman graphs of μ-bounded graphs and proving they are 2-fat line, the authors translate spectral information into combinatorial clique-structure constraints, avoiding Ramsey-theoretic arguments. These tools are then applied to the local graphs of distance-regular graphs with classical parameters, yielding explicit bounds on the parameter α in terms of b and D (notably α ≤ 2 when b=2 and D≥12) and establishing broader α-bounds under various regimes. The results illuminate the local and global structure of distance-regular graphs, offering quantitative bounds that aid in classification and characterization efforts in algebraic graph theory.
Abstract
In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $μ$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $μ$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,α,β)$. Consequently, we show that the parameter $α$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $α\leq 2$ if $b =2$ and $D \geq 12$.