Augmented k-ary n-cubes

Abstract

We define an interconnection network AQ n, k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ n, k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ n, k : is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4 n − 2 and wide-diameter at most max{( n − 1) k − ( n − 2), k + 7}; has diameter k 3 + k - 1 3 , when n = 2; and has diameter at most k 4 ( n + 1 ) , for n ⩾ 3 and k even, and at most k 4 ( n + 1 ) + n 4 , for n ⩾ 3 and k odd.