Abstract The locally twisted cube L T Q 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 L T Q n remains unclear. In this paper, we first prove that L T Q 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 L T Q n with n ≥ 4 is isomorphic to Z 2 n − 1 . These show that L T Q n with n ≥ 4 is not edge-transitive, and its full automorphism group has exactly two orbits on the vertex set of L T Q n (and consequently it is not vertex-transitive and not a Cayley graph). What is more, the symmetry of L T Q 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 L T Q n , which generalize some previous works.

中文翻译：

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局部扭曲立方体的对称性和可靠性

摘要 局部扭曲立方体LTQ n 是超立方体Q n 的变体，由Yang 等人提出。(2005) 作为并行计算的互连网络。Q n 的对称性是众所周知的，例如，它是一个边传递凯莱图。然而，LTQ n 的对称性仍不清楚。在本文中，我们首先证明 n ≥ 4 的 LTQ n 同构于阶为 2 n − 1 的初等阿贝尔 2-群 Z 2 n − 1 的双凯莱图，然后证明完全自同构群n ≥ 4 的LTQ n 与Z 2 n − 1 同构。这些表明 n ≥ 4 的 LTQ n 不是边传递的，并且它的完全自同构群在 LTQ n 的顶点集上恰好有两个轨道（因此它不是顶点传递的，也不是凯莱图）。更，LTQ n 与 n ≥ 4 的对称性也意味着它可以分解为两个顶点不相交的 ( n − 1 ) 维超立方体和一个完美匹配。作为一个应用，我们获得了 LTQ n 的 k -额外连接和 ( k + 1 ) - 分量连接，其中 k ≤ n − 1 ，这概括了之前的一些工作。