洗牌立方体的对称性质与两种变体
Symmetric properties and two variants of shuffle-cubes
AI总结：
本文证明洗牌立方体$SQ_{n}$对所有$n>2$不具有顶点传递性，并提出简化洗牌立方体$SSQ_{n}$和平衡洗牌立方体$BSQ_{n}$两种顶点传递变体，给出其路由算法并证明其具有哈密顿圈嵌入性质。
摘要：
Li等人在[Inf. Process. Lett. 77 (2001) 35--41]中提出了洗牌立方体$SQ_{n}$，作为大规模并行与分布式系统中一种有吸引力的互连网络拓扑。迄今为止，洗牌立方体的对称性质仍属未知。本文证明，对所有$n>2$，$SQ_{n}$不是顶点传递的，这在互连网络中并非理想性质。为克服这一局限，本文引入两种新颖的洗牌立方体顶点传递变体，即简化洗牌立方体$SSQ_{n}$和平衡洗牌立方体$BSQ_{n}$。随后，分别给出对所有$n>2$的$SSQ_{n}$与$BSQ_{n}$路由算法。此外，我们证明对所有$n>2$，$SSQ_{n}$和$BSQ_{n}$都具有哈密顿圈嵌入性质。最后，作为附带成果，我们修正了[IEEE Trans. Comput. 46 (1997) 484--490]中性质3的一处缺陷。
英文摘要：
Li et al. in [Inf. Process. Lett. 77 (2001) 35--41] proposed the shuffle cube $SQ_{n}$ as an attractive interconnection network topology for massive parallel and distributed systems. By far, symmetric properties of the shuffle cube remains unknown. In this paper, we show that $SQ_{n}$ is not vertex-transitive for all $n>2$, which is not an appealing property in interconnection networks. To overcome this limitation, two novel vertex-transitive variants of the shuffle-cube, namely simplified shuffle-cube $SSQ_{n}$ and balanced shuffle cube $BSQ_{n}$ are introduced. Then, routing algorithms of $SSQ_{n}$ and $BSQ_{n}$ for all $n>2$ are given respectively. Furthermore, we show that both $SSQ_{n}$ and $BSQ_{n}$ possess Hamiltonian cycle embedding for all $n>2$. Finally, as a by-product, we mend a flaw in the Property 3 in [IEEE Trans. Comput. 46 (1997) 484--490].
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