AI 中文总结
本文研究Cayley图的完美状态转移，搭建经典群同构问题与量子游走的桥梁，证明多类群的Cayley图性质，构造首个无阿贝尔表示的无限族完美状态转移Cayley图及峰值转移图族。
AI 中文摘要
我们从状态转移是图的性质而非群的性质这一视角，研究Cayley图上的完美状态转移。本文搭建了非同构群的同构Cayley图这一经典问题与图上量子游走之间的桥梁。我们证明，若群的Cayley图具有指数为2的阿贝尔子群，则在三个假设中的任意一个下，该图也是阿贝尔群的Cayley图，其中两个假设来自同构Cayley图理论。对于特殊群也有同类结论：每个阶为$p^{2n+1}$的特殊p-群，若其具有共轭闭连接集，则其每个Cayley图都是$Z_p^{2n+1}$的Cayley图。基于这些结果，我们推断所综述的六篇论文中关于二面体群、双循环群、广义二面体群、$V_{8n}$和特殊2-群的完美状态转移的所有显式构造，都是阿贝尔Cayley图的非阿贝尔表示。此外，我们证明，具有指数为2的阿贝尔子群的非阿贝尔群，若其连通Cayley图存在完美状态转移，当且仅当其阶可被4整除，且确实存在真正的非阿贝尔例子。我们证明，对于每个不小于5的奇素数幂q，Pantangi和Sin提出的$SL(2,q)$图（他们证明该图存在完美状态转移）不是任何阿贝尔群的Cayley图；据我们所知，这是第一个被证明不具有阿贝尔Cayley表示的无限族完美状态转移Cayley图。我们还构造了一个具有峰值状态转移的无限族Cayley图，并确定了每个图自同构群的所有正则子群。附录记录了顶点数不超过30的连通顶点传递完美状态转移图的普查。
英文摘要
We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.
Comments37 pages, 6 figures, 2 ancillary .csv files
