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

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 $\mu$-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 $\mu$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$. Consequently, we show that the parameter $\alpha$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $\alpha \leq 2$ if $b =2$ and $D \geq 12$.