Zhizhang Shen

On the surface area of the (n,k)-star graph

We present an explicit formula for the surface area of the (n,k)-star graph, i.e., the number of nodes at a certain distance from the identity node in the graph, by identifying the unique cycle structures associated with the nodes in the graph, deriving a distance expression in terms of such structures between the identity node of the graph and any other node, and enumerating those cycle structures satisfying the distance restriction. The above surface area derivation process can also be applied to some of the other node symmetric interconnection structures defined on the symmetric group, when the aforementioned distance expression is available.

Distance formula and shortest paths for the (n,k)-star graphs

The class of (n,k)-star graphs is a generalization of the class of star graphs. Thus a distance formula for the first class implies one for the second. In this paper, we show that the converse is also true. Another important concept is the number of shortest paths between two vertices. This problem has been solved for the star graphs. We will solve the corresponding problem for the (n,k)-star graphs.

A note on the alternating group network

The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n,n−2)-star, a member of the well-known (n,k)-stars, 1≤k≤n−1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:10.1007/s11227-009-0304-7, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n−2.

On Disjoint Shortest Paths Routing on the Hypercube

We present a routing algorithm that finds n disjoint shortest paths from the source node to n target nodes in the n -dimensional hypercube in O (n 3) = O (log3 N ) time, where N = 2 n , provided that such disjoint shortest paths exist which can be verified in O (n 5/2) time, improving the previous O (n 3 logn ) routing algorithm. In addition, the development of this algorithm also shows strong relationship between the problems of the disjoint shortest paths routing and matching.

On deriving explicit formulas of the surface areas for the arrangement graphs and some of the related graphs

We derive an explicit formula for the surface area of the arrangement graph, i.e. the number of vertices at a certain distance from the identity vertex in such a graph. We also present such formulas for the star graph, the alternating group graph, and the split-star graph, via their respective structural relationship to the arrangement graph.

On the Surface Area of the (n, k)-Star Graph

We present an explicit formula of the surface area of the (n, k)-star graphs , i.e., |{v|d(e, v) = d}|, where eis the identity node of such a graph; by identifying the cyclic structures of all the nodes in the graph, presenting a minimum routing algorithm between any node in the graph and e, and enumerating those nodes v, such that d(e, v) = d.

On the surface area of the augmented cubes

The surface area of a communication network centered at a certain vertex, i.e., the number of vertices at the same distance from this given vertex within such a network, provides an important measurement of the broadcasting and other intercommunication capabilities of this network and can find several other applications in network studies. Following a generating function approach, we derive a closed-form expression of the surface area of the recently much discussed augmented cube network and its average distance.

On the Conditional Diagnosability of Cayley Graphs Generated by 2-trees and Related Networks

In this note, we utilize existing results to derive the exact value of the conditional diagnosability for Cayley graphs generated by 2-trees, which generalize the alternating group graphs. In addition, the corresponding problem for arrangement graphs and hyper Petersen networks will also be discussed.

On the surface area of the asymmetric twisted cube

We derive a surface area result for the asymmetric twisted cube, provide closed-form expressions for such results in terms of some exemplary centers, and start to make an accurate analysis of its associated average distance measurement.

A Faster Algorithm for Finding Disjoint Ordering of Sets

Consider the problem of routing from a single source node to multiple target nodes with the additional condition that these disjoint paths be the shortest. This problem is harder than the standard one-to-many routing in that such paths do not always exist. Various sufficient and necessary conditions have been found to determine when such paths exist for some interconnection networks. And when these conditions do hold, the problem of finding such paths can be reduced to the problem of finding a disjoint ordering of sets. We study the problem of finding a disjoint ordering of sets X₁, X₂, X_s where X_i ⊆{1, 2, ···, n} and s ≤ n. We present an O(n³) algorithm for doing so, under certain conditions, thus improving the previously known O(n⁴) algorithm, and consequently, improving the corresponding one-to-many routing algorithms for finding disjoint and shortest paths.