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


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.


Volume 43, Issue 6
November and December 2017
Pages 1645-1655
  • Receive Date: 22 February 2016
  • Revise Date: 06 August 2016
  • Accept Date: 06 August 2016