Thin Q-Polynomial Distance-Regular Graphs Have Bounded c₂

In this paper, we look at distance-regular graphs with induced subgraphs K_r,t, where 1 ≤ r≤ t are integers. In particular, we show that if a distance-regular graph Γ with diameter D≥ 5 contains an induced subgraph K_2,t(t≥ 2) , then t is bounded above by a function of b1θ1+1, where θ₁ is the second largest eigenvalue of Γ. Using this bound we obtain that the intersection number c₂ of a μ-graph-regular distance-regular graph with diameter D≥ 5 and large a₁ is bounded above by a fuction of b=⌈b1θ1+1⌉. We then apply this bound to thin Q-polynomial distance-regular graphs with diameter D≥ 5 and large a₁ to show that c₂ is bounded above by a function of ⌈b1θ1+1⌉. At last, we again apply the bound to thin distance-regular graphs with classical parameters (D, b, α, β) to show that the parameter α is bounded above by a function of b1θ1+1.