Abstract
It is known that finite non-bipartite locally primitive arc-transitive graphs are normal covers of ‘basic objects’—vertex quasiprimitive ones. Praeger in (J London Math Soc 47(2):227–239, 1993) showed that a quasiprimitive action of a group G on a nonbipartite finite 2-arc transitive graph must be one of four of the eight O’Nan–Scott types. In this paper, we classify the basic locally primitive graphs where the action on vertices has O’Nan–Scott type \(\mathrm{HS}\) or HC, extending the well-known Praeger’s result about ‘basic’ 2-arc transitive graphs.
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References
Baddeley, R.: Two-arc transitive graphs and twisted wreath products. J. Algebraic Combin. 2(3), 215–237 (1993)
Conder, M., Li, C.H., Praeger, C.E.: On the Weiss conjecture for finite locally primitive graphs. Proc. Edinb. Math. Soc. 43, 129–138 (2000)
Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)
Fang, X.G., Praeger, C.E.: Finite two-arc transitive graphs admitting a Suzuki simple group. Comm. Algebra 27(8), 3727–3754 (1999)
Fang, X.G., Praeger, C.E.: Finite two-arc transitive graphs admitting a Ree simple group. Comm. Algebra 27(8), 3755–3769 (1999)
Fang, X.G., Praeger, C.E., Wang, J.: Locally primitive Cayley graphs of finite simple groups. Sci. China Ser. A 44(1), 58–66 (2001)
Fang, X.G., Praeger, C.E., Wang, J.: On the automorphism groups of Cayley graphs of finite simple groups. J. Lond. Math. Soc. (2) 66(3), 563–578 (2002)
Giudici, M., Xia, B.Z.: Vertex quasiprimitive 2-arc transitive-digraphs. Ars Math. Contemp. 14, 67–82 (2018)
Godsil, C.D.: On the full automorphism group of a graph. Combinatorica 1, 243–256 (1981)
Ivanov, A.A., Praeger, C.E.: On finite affine 2-arc transitive graphs. Europ. J. Combin. 14, 421–444 (1993)
Kleidman, P., Liebeck, M.: The Subgroup Structure of the Finite Classical Groups. London Math. Soc. Lecture Notes, Series 129, Cambridge University Press,, New York, Sydney (1990)
Li, C.H.: The finite vertex-primitive and vertex-biprimitive \(s\)-transitive graphs for \(s\ge 4\). Trans. Amer. Math. Soc. 353, 3511–3529 (2001)
Li, C.H., Lou, B.G., Pan, J.M.: Finite locally primitive abelian Cayley graphs. Sci. China Ser. A 54(4), 845–854 (2011)
Li, C.H., Pan, J.M., Ma, L.: Locally primitive graphs of prime-power order. J. Aust. Math. Soc. 86, 111–122 (2009)
Li, C. H., Seress, A.: Constructions of quasiprimitive two-arc transitive graphs of product action type. In: Finite Geometries, Groups and Computation, pp. 115–124 (2006)
Li, C.H., Ling, B., Wu, C.X.: A new family of quasiprimitive 2-arc-transitive graphs of product action type. J. Comb. Theory Ser. A 150, 152–161 (2017)
Praeger, C.E.: An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs. J. Lond. Math. Soc. 47, 227–239 (1992)
Sabidussi, G.O.: Vertex-transitive graphs. Monatsh. Math. 68, 426–438 (1964)
Weiss, R.: \(s\)-Transitive graphs. In: Algebraic Methods in Graph Theory, vol. 25, pp. 827–847. North-Holland, Amsterdam-New York (1981)
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The first author was supported by the NSFC(11861076) and the NSF of Yunnan Province (2019FB139). The second author was supported by the NSFC(11171200,11931005)
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Lou, B.G., Li, C.H. Vertex-quasiprimitive locally primitive Graphs. J Algebr Comb 55, 1279–1288 (2022). https://doi.org/10.1007/s10801-021-01094-y
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DOI: https://doi.org/10.1007/s10801-021-01094-y