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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 paralleland distributed systems. By far, symmetric properties of the shuffle cuberemains unknown. In this paper, we show that $SQ_{n}$ is not vertex-transitivefor all $n>2$, which is not an appealing property in interconnection networks.To overcome this limitation, two novel vertex-transitive variants of theshuffle-cube, namely simplified shuffle-cube $SSQ_{n}$ and balanced shufflecube $BSQ_{n}$ 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].

Symmetric properties and two variants of shuffle-cubes