Classifying a family of edge-transitive metacirculant graphs
Shu-Jiao Song1
, Cai Heng Li2
and Dianjun Wang1
1Department of Mathematical Sciences, Qinghua (Tsing Hua) University, Beijing, 100084 P. R. China
2School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, P. R. China
2School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, P. R. China
DOI: 10.1007/s10801-011-0311-7
Abstract
A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups \mathbb Z p d:\mathbb Z q \mathbb{Z}_{p^{d}}{:}\mathbb{Z}_{q} with p, q odd prime, of valency coprime to p. This characterization is then used to study an isomorphism problem regarding Cayley graphs, and to construct new families of half-arc-transitive graphs.
Pages: 497–513
Keywords: edge-transitive; metacirculant; half-arc-transitive; Cayley graph
Full Text: PDF
References
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2. Feng, Y.Q., Kwak, J.H., Xu, M.Y., Zhou, J.X.: Tetravalent half-arc-transitive graphs of order p4. Eur. J. Comb. 29, 555-567 (2008)
3. Godsil, C.D.: On the full automorphism group of a graph. Combinatorica 1, 243-256 (1981)
4. Gorenstain, D.: Finite Simple Groups. Plenum, New York (1982)
5. Jones, G.: Cyclic regular subgroups of primitive permutation groups. J. Group Theory 5(4), 403-407 (2002)
6. Kovács, I.: Classifying arc-transitive circulants. J. Algebr. Comb. 20(3), 353-358 (2004)
7. Kovács, I., Maruši\check c, D., Muzychuck, M.: On dihedrants admitting arc-regular group actions. J. Algebr. Comb. 35, 409-426 (2011)
8. Li, C.H.: On isomorphisms of connected Cayley graphs. Discrete Math. 178, 109-122 (1998)
9. Li, C.H.: On isomorphisms of connected Cayley graphs II. J. Comb. Theory, Ser. B 74, 28-34 (1998)
10. Li, C.H.: On isomorphisms of finite Cayley graphs-a survey. Discrete Math. 256, 301-334 (2002)
11. Li, C.H.: The finite primitive permutation groups containing an abelian regular subgroup. Proc. Lond. Math. Soc. 87, 725-747 (2003)
12. Li, C.H.: Permutation groups with a cyclic regular subgroup and arc transitive circulants. J. Algebr. Comb. 21, 131-136 (2005)
13. Liebeck, M., Praeger, C.E., Saxl, J.: Regular Subgroups of Primitive Permutation Groups. Mem. Am. Math. Soc. 203(952) (2010), iv+74 pp
14. Maruši\check c, D.: Recent developments in half-transitive graphs. Discrete Math. 182, 219-231 (1998)
15. Maruši\check c, D., Šparl, P.: On quartic half-arc-transitive metacirculants. J. Algebr. Comb. 28, 365-395 (2008)
16. Praeger, C.E., Xu, M.Y.: Vertex-primitive graphs of order a product of two distinct primes. J. Comb. Theory, Ser. B 59, 245-266 (1993)
17. Praeger, C.E., Wang, R.J., Xu, M.Y.: Symmetric graphs of order a product of two distinct primes. J.
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