Shilla Distance-Regular Graphs with b2 = sc2

Abstract

A Shilla graph is a distance-regular graph Γ of diameter 3 whose second eigenvalue is a = a3. A Shilla graph has intersection array {ab, (a + 1)(b − 1), b2; 1, c2, 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 b2 = c2. In the present paper, Shilla graphs with b2 = sc2, 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 b2 = sc2 and b < 170, only six admissible intersection arrays are possible. For a Q-polynomial Shilla graph with b2 = sc2, 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.