On geometric distance-regular graphs with diameter three

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