Thin Q-Polynomial Distance-Regular Graphs Have Bounded $c_2$

Abstract

In this paper, we look at distance-regular graphs with induced subgraphs $K_{r,t}$, where $1\le r\le t$ are integers. In particular, we show that if a distance-regular graph $\Gamma$ with diameter $D\ge 5$ contains an induced subgraph $K_{2,t}$ $(t\ge 2)$, then t is bounded above by a function of $\frac{b_1}{\theta _1+1}$, where $\theta _1$ is the second largest eigenvalue of $\Gamma$. Using this bound we obtain that the intersection number $c_2$ of a $\mu$-graph-regular distance-regular graph with diameter $D\ge 5$ and large $a_1$ is bounded above by a fuction of $b=\lceil \frac{b_1}{\theta _1+1}\rceil$. We then apply this bound to thin Q-polynomial distance-regular graphs with diameter $D\ge 5$ and large $a_1$ to show that $c_2$ is bounded above by a function of $\lceil \frac{b_1}{\theta _1+1}\rceil$. At last, we again apply the bound to thin distance-regular graphs with classical parameters $(D,b, \alpha , \beta )$ to show that the parameter $\alpha$ is bounded above by a function of $\frac{b_1}{\theta _1 +1}$.