A Survey on Knn Odel Graphs

Knn odel graphs of even order n and degree 1 blog 2 (n)c, W;n, are graphs which have been introduced some 25 years ago as the topology underlying a time optimal algorithm for gossiping among n nodes Knn o75]. However, they have been formally deened only 5 years ago FP94]. Since then, they have been widely studied as interconnection networks, mainly because of their good properties in terms of broadcasting and gossiping BHLP97, FR98]. In particular, Knn odel graphs of order 2 k , and of degree k, are among the three most popular families of interconnection networks in the literature, along with the hypercube of dimension k, H k Lei92], and with the recursive circulant graph G(2 k ; 4) introduced by Park and Chwa in 1994 PC94]. Indeed, those three families are commonly presented as good topologies for multicomputer networks, and are comparable since they have the same number of nodes and the same degree. In this paper, we rst survey the diierent results that exist concerning Knn odel graphs, mostly in terms of broadcasting and gossiping. We complete this survey by a study of graph-theoretical properties of the \general" Knn odel graph W;n, for any even n and 1 blog 2 (n)c. Finally, we propose a rather complete study of Knn odel graphs W k;2 k , which allows to compare this topology to the hypercube of dimension k, H k , and the recursive circulant graph G(2 k ; 4). We also provide a study of the diierent embeddings that can exist between any two of these topologies.