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Abstract: A Shilla distance-regular graph G say with valency k is a distance-regulargraph with diameter 3 such that its second largest eigenvalue equals to a3. Wewill show that a3 divides k for a Shilla distance-regular graph G, and for G wedefine b=bG:=k-a3. In this paper we will show that there are finitely manyShilla distance-regular graphs G with fixed bG>=2. Also, we will classifyShilla distance-regular graphs with bG=2 and bG=3. Furthermore, we willgive a new existence condition for distance-regular graphs, in general.

Author: Jack H. Koolen, Jongyook Park



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