On geometric distance-regular graphs with diameter three

In this paper we study distance-regular graphs with intersection array equation(1){(t+1)s,ts,(t−1)(s+1−ψ);1,2,(t+1)ψ}{(t+1)s,ts,(t−1)(s+1−ψ);1,2,(t+1)ψ} where s,t,ψs,t,ψ are integers satisfying t≥2t≥2 and 1≤ψ≤s1≤ψ≤s. Geometric distance-regular graphs with diameter three and c2=2c2=2 have such an intersection array. We first show that if a distance-regular graph with intersection array (1) exists, then ss is bounded above by a function in tt. Using this we show that for a fixed integer t≥2t≥2, there are only finitely many distance-regular graphs of order (s,t)(s,t) with smallest eigenvalue −t−1−t−1, diameter D=3D=3 and intersection number c2=2c2=2 except for Hamming graphs with diameter three. Moreover, we will show that if a distance-regular graph with intersection array (1) for t=2t=2 exists then (s,ψ)=(15,9)(s,ψ)=(15,9). As Gavrilyuk and Makhnev (2013)  [9] proved that the case (s,ψ)=(15,9)(s,ψ)=(15,9) does not exist, this enables us to finish the classification of geometric distance-regular graphs with smallest eigenvalue −3−3, diameter D≥3D≥3 and c2≥2c2≥2 which was started by the first author (Bang, 2013)  [1].