Thin Q-Polynomial Distance-Regular Graphs Have Bounded c2

초록

In this paper, we look at distance-regular graphs with induced subgraphs K-r, where 1 <= r <= t are integers. In particular, we show that if a distance-regular graph Gamma with diameter D >= 5 contains an induced subgraph K-2,K-t(t >= 2) then t is bounded above by a function of 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 >= 5 and large a(1) is bounded above by a fuction of b = left perpendiculard b(1)theta(1)+1 right perpendicular. We then apply this bound to thin Q-polynomial distance-regular graphs with diameter D >= 5 and large a(1) to show that c(2) is bounded above by a function of left perpendiculard b(1)theta(1)+1 right perpendicular. 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 left perpendiculard b(1)theta(1)+1 right perpendicular.

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