Pentavalent symmetric graphs of order twice a prime power
Pentavalent symmetric graphs of order twice a prime power
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两倍素数幂的五价对称图
DOI：
10.1016/j.disc.2016.05.008
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发表时间：
2016
影响因子：
0.8
通讯作者：
Li Yan-Tao
中科院分区：
文献类型：
--
作者：
Feng Yan-Quan;Zhou Jin-Xin;Li Yan-Tao
素数度连通对称图是基本的，如果它的自同构群不包含具有两个以上轨道的非平凡正规子群。设p为素数，n为正整数。本文研究了2pn阶连通五值对称图的性质，证明了2pn阶连通五值对称图是基本的当且仅当它是6，16,250阶图或广义二面体群上三个无限族的Cayley图的图--一个族的阶为2p且p=5或5|(p−1)，一个族的阶为2p2(p±1)，另一个族的阶为2p4，并计算了这些基本图的自同构群.关于阶为2pn的三次和四叶对称图也做了类似的工作。证明了2pn阶连通五值对称图的基本图是偶极子Dip 5的对称基本阿贝尔覆盖，并利用覆盖技术确定了这些基本图的唯一性和自同构群.此外，还对偶极子Dip5的对称Zpn-覆盖进行了分类。作为副产品，对2p2阶连通五值对称图进行了分类。
A connected symmetric graph of prime valency is basic if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let p be a prime and n a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2 p n, and it is shown that a connected pentavalent symmetric graph of order 2 p n is basic if and only if it is either a graph of order 6, 16, 250, or a graph of three infinite families of Cayley graphs on generalized dihedral groups—one family has order 2 p with p= 5 or 5∣(p− 1), one family has order 2 p 2 with 5∣(p±1), and the other family has order 2 p 4. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2 p n have been done. It is shown that basic graphs of connected pentavalent symmetric graphs of order 2 p n are symmetric elementary abelian covers of the dipole Dip 5, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric Z p n-covers of the dipole Dip 5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2 p 2 are classified.