Two-geodesic transitive graphs of prime power order

Document Type: Research Paper

Author

1 School of Statistics‎, ‎Jiangxi University of Finance and Economics‎, ‎Nanchang‎, ‎Jiangxi‎, ‎330013‎, ‎P.R‎. ‎China

2 Research Center of Applied Statistics‎, ‎Jiangxi University of Finance and Economics‎, ‎Nanchang‎, ‎Jiangxi‎, ‎330013‎, ‎P.R‎. ‎China.

Abstract

In a non-complete graph $\Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $\Gamma$ is said to be   $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power order. Next, we classify such graphs which are also vertex quasiprimitive.

Keywords

Main Subjects

References

B. Alspach, M. Conder, D. Marušič and M. Y. Xu, A classification of 2-arc-transitive circulants, J. Algebraic Combin. 5 (1996), no. 2, 83--86.

R.W. Baddeley, Two-arc transitive graphs and twisted wreath products, J. Algebraic Combin. 2 (1993), no. 3, 215--237.

A.E. Brouwer, A.M. Cohenand A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin-Heidelberg-New York, 1989.

A.E. Brouwer and H.A. Wilbrink, Ovoids and fans in the generalized quadrangle GQ(4; 2), Geom. Dedicata 36 (1990), no. 1, 121--124.

J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

A. Devillers, M. Giudici, C.H. Li and C.E. Praeger, Locally s-distance transitive graphs, J. Graph Theory 69 (2012), no. 2, 176--197.

A. Devillers, W. Jin, C.H. Li and C.E. Praeger, Local 2-geodesic transitivity and clique graphs, J. Combin. Theory Ser. A 120 (2013), no. 2, 500--508. A. Devillers, W. Jin, C.H. Li and C.E. Praeger, Line graphs and geodesic transitivity,

Ars Math. Contemp. 6 (2013) 13--20.

A. Devillers, W. Jin, C.H. Li and C.E. Praeger, On normal 2-geodesic transitive Cayley graphs, J. Algebraic Comb. 39 (2014), no. 4, 903--918.

A. Devillers, W. Jin, C.H. Li and C.E. Praeger, Finite 2-geodesic transitive graphs of prime valency, J. Graph Theory 80 (2015), no. 1, 18--27.

S.F. Du, R.J. Wang and M.Y. Xu, On the normality of Cayley digraphs of order twice a prime, Australas. J. Combin. 18 (1998) 227--234.

C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981) 243--256.

C.D. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.

R.M. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), no. 2, 304--311.

A.A. Ivanov and C.E. Praeger, On finite affine 2-arc transitive graphs, European J. Combin. 14 (1993), no. 5, 421--444.

W. Jin, Vertex quasiprimitive 2-geodesic transitive graphs, in preparation.

W. Jin, A. Devillers, C.H. Li and C.E. Praeger, On geodesic transitive graphs, Discrete Math. 338 (2015), no. 3, 168--173.

C.H. Li, Finite s-arc transitive graphs of prime power order, Bull. Lond. Math. Soc. 33 (2001), no. 2, 129--137.

C.H. Li, J.M. Pan and L. Ma, Locally primitive graphs of prime power order, J. Aust. Math. Soc. 86 (2009), no. 1, 111--122.

C.H. Li and Á. Seress, Constructions of quasiprimitive two-arc transitive graphs of product action type, in: Finite Geometries, Groups, and Computation, pp. 115--123, Walter de Gruyter, Berlin, 2006.

D. Marušič and P. Potočnik, Classifying 2-arc-transitive graphs of order a product of two primes, Discrete Math. 244 (2002), no. 1-3, 331--338.

C.E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. Lond. Math. Soc. 47 (1993), no. 2, 227--239.

C.E. Praeger, Finite normal edge-transitive Cayley graphs, Bull. Aust. Math. Soc. 60 (1999), no. 2, 207--220.

C.E. Praeger, C.H. Li and A.C. Niemeyer, Finite transitive permutation groups and finite vertex-transitive graphs, in: Graph Symmetry (Montreal, PQ, 1996), pp. 277--318,

NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 497, Kluwer Acad. Publ., Dordrecht, 1997

C.E. Praeger, J. Saxl and K. Yokohama, Distance transitive graphs and finite simple groups, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 1--21.

Á. Seress, Toward the classification of s-arc transitive graphs, in: Groups St. Andrews 2005, Vol. 2, pp. 401--414, Lond. Math. Soc. Lecture Note Ser. 340, Cambridge Univ. Press, Cambridge, 2007

W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947) 459-- 474.

W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621--624.

R. Weiss, s-transitive graphs, Colloquia Mathematica Societatis Janos Bolyai, Algebraic methods in graph theory, szeged (Hungary) 25 (1978) 827--847.

R. Weiss, The non-existence of 8-transitive graphs, Combinatorica 1 (1981) 309--311.

M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), no. 1-3, 309--319.

History

• Receive Date: 22 February 2016
• Revise Date: 06 August 2016
• Accept Date: 06 August 2016