A survey on Knödel graphs

Knodel graphs of even order n and degree 1 ≤ Δ ≤ ⌊log2(n)⌋, WΔ,n, are graphs which have been introduced some 25 years ago as the topology underlying a time optimal algorithm for gossiping among n nodes (Discrete Math. 13 (1975) 95). However, they have been formally defined only 7 years ago (Networks 38 (2001) 150). Since then, they have been widely studied as interconnection networks, mainly because of their good properties in terms of broadcasting and gossiping (Int. J. Foundations Comput. Sci. 8(2) (1997) 109, Proceedings of the 24th International Workshop on Graph-Theoretic Concepts in Computer Science, Vol. 1517, Smolenice, LNCS, 1998, p.63). In particular, Knodel graphs of order 2k, and of degree k, are among the three most popular families of interconnection networks in the literature, along with the hypercube of dimension k, Hk (Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes, Morgan Kaufman Publisher, Los Altos, CA, 1992), and with the recursive circulant graph G(2k, 4) introduced by Park and Chwa in 1994 (Proceedings of the International Symposium on Parallel Architectures, Algorithms and Networks, ISPAN'94, Kanazawa, Japan, 1994, p. 73). 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 first survey the different results that exist concerning Knodel graphs, mostly in terms of broadcasting and gossiping. We complete this survey by a study of graph-theoretical properties of the "general" Knodel graph WΔ, n, for any even n and 1 ≤ Δ ≤ ⌊log2 (n)⌋. Finally, we propose a rather complete study of Knodel graphs Wk,2k, which allows to compare this topology to the hypercube of dimension k, Hk, and the recursive circulant graph G(2k, 4). We also provide a study of the different embeddings that can exist between any two of these topologies.