1-Homogeneous Graphs with Cocktail Party -Graphs

Let be a graph with diameter d ≥ 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition {{z ∈ V ( ) | ∂(z, y) = i, ∂(x, z) = j} | 0 ≤ i, j ≤ d} is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v′, k′, λ′, μ′), where v′ = k, k′ = a1, (v′ − k′ − 1)μ′ = k′(k′ − 1 − λ′) and c2 ≥ μ′ + 1, since a μ-graph is a regular graph with valency μ′. If c2 = μ′ + 1 and c2 = 1, then is a Terwilliger graph, i.e., all the μ-graphs of are complete. In [11] we classified the Terwilliger 1homogeneous graphs with c2 ≥ 2 and obtained that there are only three such examples. In this article we consider the case c2 = μ′ + 2 ≥ 3, i.e., the case when the μ-graphs of are the Cocktail Party graphs, and obtain that either λ′ = 0, μ′ = 2 or is one of the following graphs: (i) a Johnson graph J (2m, m) with m ≥ 2, (ii) a folded Johnson graph J̄ (4m, 2m) with m ≥ 3, (iii) a halved m-cube with m ≥ 4, (iv) a folded halved (2m)-cube with m ≥ 5, (v) a Cocktail Party graph Km×2 with m ≥ 3, (vi) the Schläfli graph, (vii) the Gosset graph.