On almost good triples of vertices in edge regular graphs

Abstract

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b1 = k−λ−1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k − 4b1 + 3 (and µ(u, w) + µ(u, z) = 2k − 4b1 + 4). If k = 3b1 + γ with γ ≥ −2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b1 = 6, and v = 31; or Δ is a 4-clique and Γ is a Schlafli graph.