Kautz

Kautz is a surname of German origin that is found in various countries and disciplines. It is associated with individuals in science, arts, and public life, among other fields. In addition to its use as a family name, Kautz appears in technical literature as the designation of a specific mathematical construct and related concepts.

In mathematics and computer science, the Kautz graph is a class of directed graphs used as interconnection networks for parallel computing and related applications. A Kautz graph is defined for integers q ≥ 1 and n ≥ 1 using an alphabet of size q+1. The vertices are all sequences a1 a2 ... an where each ai is in {0,1,...,q} and consecutive symbols differ (that is, ai ≠ ai+1 for all i). There is a directed edge from a1 a2 ... an to a2 a3 ... an b whenever b ∈ {0,1,...,q} and b ≠ an. This construction yields a regular graph with out-degree q and in-degree q, and the number of vertices is (q+1) q^{n-1}. The diameter of a Kautz graph is n−1, giving it a small worst-case path length relative to the number of vertices. Kautz graphs are valued in both theoretical studies and practical network design due to their high connectivity, sparse degree, and favorable diameter properties, making them suitable as models for interconnect topologies in multiprocessor systems.

See also: de Bruijn graph.