Matrix-Star Graphs: A New Interconnection Network Based on Matrix Operations

Abstract

In this paper, we introduce new interconnection networks matrix-star graphs MTS _n1,...,nk where a node is represented by n ₁ × ... × n _k matrix and an edge is defined by using matrix operations. A matrix-star graph MTS _2,n can be viewed as a generalization of the well-known star graph such as degree, connectivity, scalability, routing, diameter, and broadcasting. Next, we generalize MTS _2,n to 2-dimensional and 3-dimensional matrix-star graphs MTS _{k, n} , MTS _{k, n,p} . One of important desirable properties of interconnection networks is network cost which is defined by degree times diameter. The star graph, which is one of popular interconnection topologies, has smaller network cost than other networks. Recently introduced network, the macro-star graph has smaller network cost than the star graph. We further improve network cost of the macro-star graph: Comparing a matrix-star graph $MTS_{k,k,k}(k = \sqrt[3]{n^{2}})$ with n ²! nodes to a macro-star graph MS(n–1,n–1) with ((n–1)²+1)! nodes, network cost of MTS _{k,k, k} is O(n ^2.7) and that of MS(n–1,n–1) is O(n ³). It means that a matrix-star graph is better than a star graph and a macro-star graph in terms of network cost.