Codes in Shilla Distance-Regular Graphs

Abstract

Let Γ be a distance-regular graph of diameter 3 containing a maximal 1-code C, which is locally regular and last subconstituent perfect. Then the graph Γ has intersection array {a(p + 1), cp, a + 1; 1,c, ap} or {a(p + 1), (a + 1)p, c;1, c, ap}, where a = a3, c = c2, and p = p 33 3 (Jurisic, Vidali). In the first case, Γ has eigenvalue θ2 = −1 and the graph Γ3 is pseudogeometric for GQ(p + 1, a). In the second case, Γ is a Shilla graph. We study Shilla graphs in which every two vertices at distance 2 belong to a maximal 1-code. It is proved that, in the case θ2 = −1, a graph with the specified property is either the Hamming graph H(3, 3) or a Johnson graph. We find necessary conditions for the existence of Q-polynomial Shilla graphs in which any two vertices at distance 3 lie in a maximal 1-code. In particular, we find two infinite families of feasible intersection arrays of Q-polynomial graphs with the specified property: {b(b2 − 3b)/2, (b − 2)(b − 1)2/2, (b − 2)t/2; 1, bt/2, (b2 − 3b)(b − 1)/2} (graphs with p 33 3 = 0) and {b2(b − 4)/2, (b2 − 4b + 2)(b − 1)/2, (b − 2)l/2; 1, bl/2, (b2 − 4b)(b − 1)/2} (graphs with p 33 3 = 1).