Super connectivity of k-regular interconnection networks

Super connectivity is an important issue in interconnection networks. It has been shown that if a network possesses the super connectivity property, it has a high reliability and a small vertex failure rate. Many interconnection networks, like the hypercubes, twisted-cubes, crossed-cubes, möbius cubes, split-stars, and recursive circulant graphs, are proven to be super connected; and the augmented cubes are maximum connected. However, each network vertex has a higher degree as long as the number of vertices increases exponentially. For example, each vertex of the hypercube Qn has a degree of n, and each vertex of the augmented cube AQn has a degree of 2n − 1. In this paper, we not only show that the augmented cube AQn is super connected for n = 1, 2 and n ⩾ 4, but also propose a variation of AQn, denoted by AQn,i, such that V(AQn,i) =  V(AQn), E(AQn,i) ⊆ E(AQn), and AQn,i is i-regular with n ⩾ 3 and 3 ⩽ i ⩽ 2n − 1, in which AQn,i is also super connected. In addition, we state the diameter of AQn,i.