On symmetric graphs of valency five
On symmetric graphs of valency five
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DOI：
10.1016/j.disc.2009.11.019
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发表时间：
2010-06
期刊：
影响因子：
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通讯作者：
Jin-Xin Zhou;Yan-Quan Feng
中科院分区：
文献类型：
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作者：
Jin-Xin Zhou;Yan-Quan Feng
一个图X，其子群G是X的自同构群Aut（X）的子群，如果G在s-弧上可迁而在（s+1）-弧上不可迁，则称X是（G，s）-可迁的，对某些s≥1，如果G是（Aut（X），s）-可迁的.设X是连通（G，s）-传递图，G_v是G中顶点v∈V（X）的稳定子.若X的化合价为5且Gvis可解，则韦斯[R.M.韦斯，p-因子分解方法在对称图中的应用，数学学报。腓Soc.85（1979）43-48]证明了s≤3，本文证明了Gvis同构于循环群Z5，二面体群D10或D20（s=1），Frobenius群F20或F20× Z2（s=2），F20× Z4（s=3）.进一步证明了对于非交换单群G上的5度连通1-传递Cayley图Cay（G，S），Cay（G，S）的自同构群是半直积R（G）<$Aut（G，S），其中R（G）是G的右正则表示，Aut（G，S）={α∈Aut（G）<$Sα=S}.
A graph X, with a subgroup G of the automorphism group Aut(X) of X, is said to be (G,s)-transitive, for some s≥1, if G is transitive on s-arcs but not on (s+1)-arcs, and s-transitive if it is (Aut(X),s)-transitive. Let X be a connected (G,s)-transitive graph, and Gvthe stabilizer of a vertex v∈V(X) in G. If X has valency 5 and Gvis solvable, Weiss [R.M. Weiss, An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43–48] proved that s≤3, and in this paper we prove that Gvis isomorphic to the cyclic group Z5, the dihedral group D10or the dihedral group D20for s=1, the Frobenius group F20or F20×Z2for s=2, or F20×Z4for s=3. Furthermore, it is shown that for a connected 1-transitive Cayley graph Cay(G,S) of valency 5 on a non-abelian simple group G, the automorphism group of Cay(G,S) is the semidirect product R(G)⋊Aut(G,S), where R(G) is the right regular representation of G and Aut(G,S)={α∈Aut(G)∣Sα=S}.