Abstract
Let Γ be a graph and G be a 2-arc transitive automorphism group of Γ. For a vertex x ∈ Γ let G(x)Γ(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set Γ(x) of vertices adjacent to x in Γ. Then Γ is said to be a locally projective graph of type (n,q) if G(x)Γ(x) contains PSLn(q) as a normal subgroup in its natural doubly transitive action. Suppose that Γ 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 Γ(x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n=4, q=2, Γ has 506 vertices and \(G \cong M_{23,} {\text{or }}q = 4,PSL_n (4) \leqslant G(x) \leqslant PGL_n (4) \), and Γ 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.
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Ivanov, A., Praeger, C.E. On Locally Projective Graphs of Girth 5. Journal of Algebraic Combinatorics 7, 259–283 (1998). https://doi.org/10.1023/A:1008619222465
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DOI: https://doi.org/10.1023/A:1008619222465