Delsarte Set Graphs with Small c(2)

Abstract

Let Gamma be a Delsarte set graph with an intersection number c(2) (i.e., a distance-regular graph with a set C of Delsarte cliques such that each edge lies in a positive constant number n(C) of Delsarte cliques in C). We showed in Bang et al. (J Combin 28:501-506, 2007) that if psi(1) > 1 then c(2) >= 2 psi(1) where psi(1) := vertical bar Gamma(1)(x)boolean AND C vertical bar for x is an element of V(Gamma) and C aDelsarte clique satisfying d(x, C) = 1. In this paper, we classify Gamma with the case c(2) = 2 psi(1) > 2. As a consequence of this result, we show that if c(2) <= 5 and psi(1) > 1 then Gamma is either a Johnson graph or a folded Johnson graph (J) over bar (4s, 2s) with s >= 3.