In this paper, we undertake a comparative study of two important interconnection network topologies: the star graph and the hypercube, from the graph theory point of view. Topological properties are derived for the star graph and are compared with the corresponding properties of the hypercube. Among other results, we determine necessary and sdcient conditions for shortest path routing and we characterize maximum-sized families of parallel paths between any two nodes of the star graph. These parallel paths are proven of minimum length within a small additive constant. We also define greedy and asymptotically balanced spanning trees to support broadcasting and personalized communication on the star graph. These results confirm the already claimed topological superiority of the star graph over the hypercube.

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We obtain the fault diameter of k-ary n-cube interconnection networks (also known as n-dimensional k-torus networks). We start by constructing a complete set of node-disjoint paths (i.e., as many paths as the degree) between any two nodes of a k-ary n-cube. Each of the obtained paths is of length zero, two, or four plus the minimum length except for one path in a special case (when the Hamming distance between the two nodes is one) where the increase over the minimum length may attain eight. These results improve those obtained in [8] where the length of some of the paths has a variable increase (which can be arbitrarily large) over the minimum length. These results are then used to derive the fault diameter of the k-ary n-cube, which is shown to be D + 1 where D is the fault free diameter.

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We study the cross product as a method for generating and analyzing interconnection network topologies for multiprocessor systems. Consider two interconnection graphs G 1 and G 2 each with some established properties such as symmetry, low degree and diameter, scalability, simple optimal routing, recursive structure (partitionability), fault tolerance, existence of nodedisjoint paths, low cost embedding, and efficient broadcasting. We investigate and evaluate the corresponding properties for the cross product of G 1 and G 2 based on the properties of G 1 and those of G 2. We also give a mathematical characterization of product families of graphs which are closed under the cross product operation. This investigation is useful in two ways. On one hand, it gives a new tool for further studying some of the known interconnection topologies, such as the hypercube and the mesh, which can be defined using the cross product operation. On the other hand, it can be used in defining and evaluating new interconnection graphs using the cross product operation on known topologies.

ABSTRACT. Let Tn(f) = (OjA^-q be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function. An identity is developed for det(Tn(f) -A) which may be used to prove that the limit set of the eigenvalues of the Tn(f) is a point or consists of a finite number of analytic arcs.

This paper derives the conditional node connectivity of the k-ary n-cube interconnection network under the condition of forbidden faulty sets (i.e. assuming that each non-faulty processor has at least one non-faulty neighbor). It is shown that under this condition and for k≥4 and n≥2, the k-ary n-cube, whose connectivity is 2n, can tolerate up to 4n-3 faulty nodes without becoming disconnected. The conditional node connectivity in this case is therefore 4n-2. For k=3 and n≥2 the established conditional node connectivity is 4n-3. The result for the remaining smaller values of k and n are also obtained.

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