On locally projective graphs of girth 5 (original) (raw)

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x ∈ let G(x) (x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n, q) if G(x) (x) contains PSL n (q) as a normal subgroup in its natural doubly transitive action. Suppose that 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 (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n = 4, q = 2, has 506 vertices and G ∼ = M 23 , or q = 4, PSL n (4) ≤ G(x) ≤ PGL n (4), and 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 2 20 vertices.