Interconnection networks and their eigenvalues

Abstract

Interconnection networks of various topologies are used in parallel computing. It is important to study the graph theoretical/combinatorial properties of the underlying networks in order to better understand them and develop more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms. In this paper, we approach this problem from a new angle by looking into the spectra (eigenvalues and their multiplicities) of these networks. Eigenvalues of the adjacency matrix of a graph can reveal certain properties of the graph since they are closely related to some of its combinatorial invariants. Specifically, for some of the popular interconnection networks, we study their eigenvalues and multiplicities by (1) summarizing the currently available results; (2) deriving some of these results in a more straightforward way; (3) obtaining new results; and (4) presenting experimental results on several interconnection networks. In addition, we briefly survey the results that relate spectra of graphs to their structural properties. Although much work remains to be done, by looking into the spectra of interconnection networks, we hope to bring about a more unified approach to studying their topological properties.