Augmented k-ary n-cubes

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We define an interconnection network AQ n,k which we call the augmented kary 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 4n−2 and wide-diameter at most max{(n−1)k−(n−2), k+7}; has diameter ⌊ k 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.

A kind of conditional vertex connectivity of Cayley graphs generated by 2-trees

“…Hypercubes and its variants [9,15,17,21,22,24,25,28,36,37] form the basic classes of interconnection networks. The class of star graphs [1] and the class of alternating group graphs [23] were introduced to be competitive models of hypercubes.…”

A general algorithm for detecting faults under the comparison diagnosis model

“…A graph called the augmented k-ary n-cube AQ n,k was recently defined in [25] and is an extension of the k-ary n-cube in a manner analogous to the extension of an n-dimensional hypercube to an n-dimensional augmented cube. It was proven in [25] that AQ n,k is regular of degree 4n − 2 and has connectivity 4n − 2.…”

“…It was proven in [25] that AQ n,k is regular of degree 4n − 2 and has connectivity 4n − 2. Thus, by [6], so long as (n, k) = (2, 3), AQ n,k has diagnosability 4n − 2.…”

Topological Properties on the Wide and Fault Diameters of Exchanged Hypercubes

“…Moreover, this interconnection network has many benefits such as parallel routing and fault tolerance. In recent years, many literature references discuss the topic of internally disjoint paths in some specific networks, such as hypercubes [20], crossed cubes [8], ðn; kÞ-star graphs [12], folded hypercubes [15], hypercubelike graphs [19], hierarchical hypercubes [21], and augmented k-ary n-cubes [22]. Next, we discuss the fault and wide diameters of exchanged hypercubes.…”

“…The fault diameter, which was first proposed in [9], is used to estimate the effects of faults on the diameter, while the wide diameter is used to measure the diameter of the connections with prescribed bandwidths, and it is a combination of both the diameter and connectivity. The fault and wide diameters have been discussed in [2], [3], [4], [6], [12], [15], [21], [22]. In this study, we construct s þ 1 (or t þ 1) internally disjoint paths between any two vertices for parallel routes in the exchanged hypercube EHðs; tÞ for 3 s t. We also prove that both the ðs þ 1Þ-wide diameter and s-fault diameter are s þ t þ 3 for 3 s t.…”