Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T23:26:13.400Z Has data issue: false hasContentIssue false
  • English
  • Français

FINITE 3-GEODESIC TRANSITIVE BUT NOT 3-ARC TRANSITIVE GRAPHS

Part of: Permutation groups Algebraic combinatorics

Published online by Cambridge University Press:  23 September 2014

WEI JIN
WEI JIN*
Affiliation:
School of Statistics, Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, PR China email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we first prove that for $g\in \{3,4\}$, there are infinitely many 3-geodesic transitive but not 3-arc transitive graphs of girth $g$ with arbitrarily large diameter and valency. Then we classify the family of 3-geodesic transitive but not 3-arc transitive graphs of valency 3 and those of valency 4 and girth 4.

Keywords

3-geodesic transitive graph3-arc transitive graphautomorphism group

MSC classification

Primary: 05E18: Group actions on combinatorial structures
Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 183 - 190
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Alspach, B., Conder, M., Marušič, D. and Xu, M. Y., ‘A classification of 2-arc-transitive circulants’, J. Algebraic Combin. 5 (1996), 83–86.Google Scholar
Brouwer, A. E., Cohen, A. M. and Neumaier, A., Distance-Regular Graphs (Springer, Berlin, 1989).CrossRefGoogle Scholar
Devillers, A., Jin, W., Li, C. H. and Praeger, C. E., ‘Local 2-geodesic transitivity and clique graphs’, J. Combin. Theory Ser. A 120 (2013), 500–508.CrossRefGoogle Scholar
Devillers, A., Jin, W., Li, C. H. and Praeger, C. E., ‘Line graphs and geodesic transitivity’, Ars Math. Contemp. 6 (2013), 13–20.CrossRefGoogle Scholar
Devillers, A., Jin, W., Li, C. H. and Praeger, C. E., ‘On normal 2-geodesic transitive Cayley graphs’, J. Algebraic. Combin. 39 (2014), 903–918.Google Scholar
Dixon, J. D. and Mortimer, B., Permutation Groups (Springer, New York, 1996).Google Scholar
Ivanov, A. A. and Praeger, C. E., ‘On finite affine 2-arc transitive graphs’, European J. Combin. 14 (1993), 421–444.Google Scholar
Li, C. H. and Pan, J. M., ‘Finite 2-arc-transitive abelian Cayley graphs’, European J. Combin. 29 (2008), 148–158.Google Scholar
Praeger, C. E., ‘An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs’, J. London Math. Soc. 47(2) (1993), 227–239.Google Scholar
Tutte, W. T., ‘A family of cubical graphs’, Proc. Cambridge Philos. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
Tutte, W. T., ‘On the symmetry of cubic graphs’, Canad. J. Math. 11 (1959), 621–624.Google Scholar
Weiss, R., ‘The non-existence of 8-transitive graphs’, Combinatorica 1 (1981), 309–311.Google Scholar