On 2-transitive graphs of girth 5
A graph \(\Gamma\) is said to be 2-transitive if its automorphism group G acts transitively on the set of 2-paths of \(\Gamma\) but intransitively on the set of 3-paths. Suppose that the 2-transitive graph \(\Gamma\) satisfies: (1) for any vertex x, the action of the stabilizer G(x) on the set of neighbors of x is doubly-transitive with no regular normal subgroup, and (2) the subgroup of G(x) fixing pointwise every neighbor of x is nontrivial. \textit{C. E. Praeger} [Combinat. Math. IX, Proc. 9th Austr. Conf. 1981, Lect. Notes Math. 952, 123-141 (1982; Zbl 0494.05028)] claimed that \(\Gamma\) must have girth less than 5. The author constructs finite and infinite counterexamples, all having girth 5. In the finite counterexamples, possibilities for G include \(M_{22}\), \(Co_ 2\), \(J_ 4\), and Fischer's group \(F_ 2.\) In a self-contained subinvestigation, it is shown that a graph of girth 5 in which every pair of vertices lying at distance 2 apart belong to exactly m 5-circuits, while no 6-circuit contains only shortest paths, contains \((m+1)\)-valent Moore graphs of diameter 2, where \(m\in \{1,2,6,56\}.\) This paper requires familiarity with the classification of finite group, combinatorial designs, and codes in addition to algebraic graph theory. Unfortunately the author's unfamiliarity with English syntax, usage, and grammar, together with occasional mathematical misprints, render it difficult to read.
scientific article; zbMATH DE number 4051665
scientific article