On Locally Projective Graphs of Girth 5

Let0 be a graph and G be a 2-arc transitive automorphism group of0. For a vertex x ∈ 0 let G(x)0(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set 0(x) of vertices adjacent to x in 0. Then 0 is said to be a locally projective graph of type (n, q) if G(x)0(x) contains PSLn(q) as a normal subgroup in its natural doubly transitive action. Suppose that 0 is a locally projective graph of type (n, q), for some n ≥ 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on 0(x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n = 4, q = 2, 0 has 506 vertices and G ∼= M23, or q = 4, PSLn(4) ≤ G(x) ≤ PGLn(4), and 0 contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W (n) of which all other graphs for this n are quotients. The graph W (3) satisfies the conditions and has 220 vertices.