Abstract
A graph is called edge-primitive if its automorphism group acts primitively on its edge-set. In this paper, edge-primitive graphs of prime power order are determined.
Similar content being viewed by others
References
Cameron, P.J.: Finite permutation groups and finite simple groups. Bull. Lond. Math. Soc. 13, 1–22 (1981)
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, London (1985)
Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)
Giudici, M., Li, C.H.: On finite edge-primitive and edge-quasiprimitive graphs. J. Comb. Theory Ser. B 100, 275–298 (2009)
Guo, S.T., Feng, Y.Q., Li, C.H.: Edge-primitive tetravalent graphs. J. Comb. Theory Ser. B 112, 124–137 (2015)
Guo, S.T., Feng, Y.Q., Li, C.H.: The finite edge-primitive pentavalent graphs. J. Algebr. Comb. 38, 491–497 (2013)
Guralnick, R.M.: Subgroups of prime power index in a simple group. J. Algebra 225, 304–311 (1983)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Li, C.H., Praeger, C.E., Venkatesh, A., Zhou, S.M.: Finite locally-quasiprimitive graphs. Discrete Math. 246, 197–218 (2002)
Li, C.H.: The finite primitive permutation groups containing an abelian regular subgroup. Proc. Lond. Math. Soc. 87, 725–748 (2003)
Li, C.H., Lou, B.G., Pan, J.M.: Finite locally primitive abelian Cayley graphs. Sci. China 54(4), 854–945 (2011)
Li, C.H., Zhang, H.: The finite primitive groups with soluble stabilizers, and the edge-primitive s-arc transitive graphs. Proc. Lond. Math. Soc. 103, 441–472 (2011)
Li, C.H., Ma, L.: Locally primitive graphs and bidirect producs of graphs. J. Aust. Math. Soc. 91(2), 231–242 (2011)
Pan, J.M., Wu, C.X., Yin, F.G.: Finite edge-primitive graphs of prime valency. Eur. J. Comb. 73, 61–71 (2018)
Pan, J. M., Wu, C. X., Yin, F. G.: Edge-primitive Cayley graphs on abelian and dihedral groups. Discrete Math. (in press)
Praeger, C.E.: Primitive permutation groups with a doubly transitive subconstituent. J. Aust. Math. Soc. Ser. A 45, 66–77 (1988)
Praeger, C.E.: The inclusion problem for finite primitive permutation groups. Proc. Lond. Math. Soc 60(3), 68–88 (1990)
Praeger, C.E.: Finite transitive permutation groups and vertex-transitive graphs. In: Hahn, G., Sabidussi, G. (eds.) Graph Symmetry: Algebraic Methods and Applications. NATO Advanced Science Institute Series C 497, pp. 277–318. Klumer, Dordrecht (1997)
Pablo, S.: Finite primitive groups and edge-transitive hypergraphs. J. Algebr. Comb. 43(3), 715–734 (2016)
Weiss, R.M.: Kantenprimitive Graphen vom Grad drei. J. Comb. Theory Ser. B 15, 269–288 (1973)
Wu, C. X., Pan, J. M.: Finite hexavalant edge-primitive graphs (submitted)
Acknowledgements
The authors are very grateful to the referees for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was partially supported by National Natural Science Foundation of China (11461007, 11231008, ZR2018PA005).
Rights and permissions
About this article
Cite this article
Pan, J., Huang, Z. & Wu, C. Edge-Primitive Graphs of Prime Power Order. Graphs and Combinatorics 35, 249–259 (2019). https://doi.org/10.1007/s00373-018-1997-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-018-1997-2