Shilla Distance-Regular Graphs with b2 = sc2

A Shilla graph is a distance-regular graph Γ of diameter 3 whose second eigenvalue is a = a 3 . A Shilla graph has intersection array { ab , ( a + 1)( b − 1), b 2 ; 1, c 2 , a ( b − 1)}. J. Koolen and J. Park showed that, for a given number b , there exist only finitely many Shilla graphs. They also found all possible admissible intersection arrays of Shilla graphs for b ∈ {2, 3}. Earlier the author together with A. A. Makhnev studied Shilla graphs with b 2 = c 2 . In the present paper, Shilla graphs with b 2 = sc 2 , where s is an integer greater than 1, are studied. For Shilla graphs satisfying this condition and such that their second nonprincipal eigenvalue is −1, five infinite series of admissible intersection arrays are found. It is shown that, in the case of Shilla graphs without triangles in which b 2 = sc 2 and b < 170, only six admissible intersection arrays are possible. For a Q -polynomial Shilla graph with b 2 = sc 2 , admissible intersection arrays are found in the cases b = 4 and 5, and this result is used to obtain a list of admissible intersection arrays of Shilla graphs for b ∈ {4, 5} in the general case.