Symmetric property and reliability of locally twisted cubes

The locally twisted cube LTQ(n) is a variant of the hypercube Q(n), which was introduced by Yang et al. (2005) as an interconnection network for parallel computing. The symmetry of Q(n) is well-known, for example, it is an edge-transitive Cayley graph. However, the symmetry of LTQ(n) remains unclear. In this paper, we first prove that LTQ(n) with n >= 4 is isomorphic to a bi-Cayley graph of an elementary abelian 2-group Z(2)(n)(-1) of order 2(n-1), and then prove that the full automorphism group of LTQ(n) with n >= 4 is isomorphic to Z(2)(n-1). These show that LTQ(n) with n >= 4 is not edge-transitive, and its full automorphism group has exactly two orbits on the vertex set of LTQ(n) (and consequently it is not vertex-transitive and not a Cayley graph). What is more, the symmetry of LTQ(n) with n >= 4 also implies that it can be decomposed to two vertex-disjoint (n- 1)-dimensional hypercubes and a perfect matching. As an application, we obtain the k-extra connectivity and (k 1)-component connectivity with k <= n - 1 of LTQ(n), which generalize some previous works. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates the symmetry of the locally twisted cube LTQ(n), proving that it is different from the hypercube Q(n). The results obtained in this study provide insights into the structure of LTQ(n) and generalize some previous works on this topic.