Graph Constructions Derived from Interconnection Networks

A class of interconnection networks for massively parallel processors are designed by taking copies of a building block network and wiring them together. For Dragonfly networks, the building block network is a complete graph and the wiring together is done by either a cycle or a complete graph. The process may be viewed as a way to construct a new graph from two component graphs. The resulting graph is known as a replacement graph. Furthermore, one of these constructions leads to a very large number of graphs, some of which are provably not isomorphic. The point is that the construction of a replacement G by H requires that G be converted to a network. This paper explains the way the graph of an interconnection network is labeled and a table which is analogous to the adjacency matrix of a labeled graph. The table is used to demonstrate the nondeterminism of the concept of a replacement graph. The graph constructions are presented along with the motivating interconnection networks. The graph constructions can be generalized.