Two-arc-transitive bicirculants. (English) Zbl 1521.05064
Summary: In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected 2-arc-transitive bicirculant is one of the following graphs: \( C_{2 n}\) where \(n \geqslant 2\), \(\operatorname{K}_{2 n}\) where \(n \geqslant 2\), \(\operatorname{K}_{n , n}\) where \(n \geqslant 3\), \(\operatorname{K}_{n , n} - n \operatorname{K}_2\) where \(n \geqslant 4\), \(B( \operatorname{PG}(d - 1, q))\) and \(B^\prime( \operatorname{PG}(d - 1, q))\) where \(d \geq 3\) and \(q\) is a prime power, \( X_1(4, q)\) where \(q \equiv 3 \pmod{4}\) is a prime power, \( \operatorname{K}_{q + 1}^{2 d}\) where \(q\) is an odd prime power and \(d \geq 2\) dividing \(q - 1\), \(A T_Q(1 + q, 2 d)\) where \(d | q - 1\) and \(d \nmid \frac{ 1}{ 2}(q - 1)\), \(A T_D(1 + q, 2 d)\) where \(d | \frac{ 1}{ 2}(q - 1)\) and \(d \geq 2\), \(\Gamma(d, q, r)\), where \(d \geq 2\), \(q\) is a prime power and \(r | q - 1\), Petersen graph, Desargues graph, dodecahedron graph, folded 5-cube, \(X(2, 2)\), \(X^\prime(3, 2)\), \(X_2(3)\), \(A T_Q(4, 12)\), \(G P(12, 5)\), \(G P(24, 5)\), \(B(H(11))\), \(B^\prime(H(11))\), \(A T_D(4, 6)\) and \(A T_D(5, 6)\).
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 05E30 | Association schemes, strongly regular graphs |
Software:
MagmaReferences:
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