Pentavalent symmetric graphs of order twice a prime power

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• 作者：Yan-Quan Feng^a ; ^{yqfeng@bjtu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jin-Xin Zhou^a ; ^{jxzhou@bjtu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yan-Tao Li^b ; ^{yantao@ygi.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Symmetric graph ; Cayley graph ; Regular covering ; Normal cover
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 November 2016
• 年：2016
• 卷：339
• 期：11
• 页码：2640-2651
• 全文大小：520 K

文摘

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$p$ be a prime and n$n$ a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2pⁿ$2p^n$, and it is shown that a connected pentavalent symmetric graph of order 2pⁿ$2p^n$ is basic if and only if it is either a graph of order aba80b8150c32cfe18e5676fd03bd485" title="Click to view the MathML source">6$6$, 16$16$, abadc6265152d5223a2681" title="Click to view the MathML source">250$250$, or a graph of three infinite families of Cayley graphs on generalized dihedral groups—one family has order 2p$2p$ with p=5$p=5$ or 5∣(p−1)$5\mid (p-1)$, one family has order 2p²$2p^2$ with 5∣(p±1)$5\mid (p\pm 1)$, and the other family has order 2p⁴$2p^4$. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2pⁿ$2p^n$ have been done.

It is shown that basic graphs of connected pentavalent symmetric graphs of order 2pⁿ$2p^n$ are symmetric elementary abelian covers of the dipole Dip₅${\text{Dip}}_5$, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric ${\mathbb{Z}}_p^n$-covers of the dipole Dip₅${\text{Dip}}_5$ are classified. As a byproduct, connected pentavalent symmetric graphs of order 2p²$2p^2$ are classified.