Automorphisms of Strongly Regular Krein Graphs without Triangles

A strongly regular graph is called a Krein graph if, in one of the Krein conditions, an equality obtains for it. A strongly regular Krein graph Kre(r) without triangles has parameters ((r² + 3r)², r³ + 3r² + r, 0, r² + r). It is known that Kre(1) is a Klebsh graph, Kre(2) is a Higman-Sims graph, and that a graph of type Kre(3) does not exist. Let G be the automorphism group of a hypothetical graph Γ = Kre(5), g be an element of odd prime order p in G, and Ω = Fix(g). It is proved that either Ω is the empty graph and p = 5, or Ω is a one-vertex graph and p = 41, or Ω is a 2-clique and p = 17, or Ω is the complete bipartite graph K_8,8, from which the maximal matching is removed, and p = 3.Supported by RFBR grant No. 05-01-00046.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 335–354, May–June, 2005.