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| [[File:Kautz graph 32 33.svg|right|400px|thumb|Example of '''Kautz graph''' on 3 characters with string length 2 (on the left) and 3 (on the right); the edges on the left correspond to the vertices on the right.]]
| | Marvella is what you can call her but it's not the most female name out there. California is our beginning place. Bookkeeping is what I do. The favorite pastime for my children and me is to perform baseball but I haven't made a dime with it.<br><br>My page ... [http://netwk.hannam.ac.kr/xe/data_2/38191 at home std testing] |
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| The '''Kautz graph''' <math>K_M^{N + 1}</math> is a
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| [[directed graph]] of degree <math>M</math> and dimension <math>N+
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| 1</math>, which has <math>(M +1)M^{N}</math> vertices labeled by all
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| possible strings <math>s_0 \cdots s_N</math> of length <math>N +
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| 1</math> which are composed of characters <math>s_i</math> chosen from
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| an alphabet <math>A</math> containing <math>M + 1</math> distinct
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| symbols, subject to the condition that adjacent characters in the
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| string cannot be equal (<math>s_i \neq s_{i+ 1}</math>).
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| The Kautz graph <math>K_M^{N + 1}</math> has <math>(M + 1)M^{N | |
| + 1}</math> edges
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| <math>\{(s_0 s_1 \cdots s_N,s_1 s_2 \cdots s_N s_{N + 1})| \; s_i \in A \; s_i \neq s_{i + 1} \} \, </math>
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| It is natural to label each such edge of <math>K_M^{N + 1}</math>
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| as <math>s_0s_1 \cdots s_{N + 1}</math>, giving a one-to-one correspondence
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| between edges of the Kautz graph <math>K_M^{N + 1}</math>
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| and vertices of the Kautz graph
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| <math>K_M^{N + 2}</math>.
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| Kautz graphs are closely related to [[De Bruijn graph]]s.
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| == Properties ==
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| * For a fixed degree <math>M</math> and number of vertices <math>V = (M + 1) M^N</math>, the Kautz graph has the smallest [[Distance_(graph_theory)|diameter]] of any possible directed graph with <math>V</math> vertices and degree <math>M</math>.
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| * All Kautz graphs have [[Eulerian cycle]]s. (An Eulerian cycle is one which visits each edge exactly once-- This result follows because Kautz graphs have in-degree equal to out-degree for each node)
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| * All Kautz graphs have a [[Hamiltonian cycle]] (This result follows from the correspondence described above between edges of the Kautz graph <math>K_M^{N}</math> and vertices of the Kautz graph <math>K_M^{N + 1}</math>; a Hamiltonian cycle on <math>K_M^{N + 1}</math> is given by an Eulerian cycle on <math>K_M^{N}</math>)
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| * A degree-<math>k</math> Kautz graph has <math>k</math> disjoint paths from any node <math>x</math> to any other node <math>y</math>.
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| == In computing ==
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| The Kautz graph has been used as a [[network topology]] for connecting processors in [[high-performance computing]]<ref>{{cite web | url=http://pl.atyp.us/wordpress/?p=1275 | title=The Kautz Graph | author=Darcy, Jeff | date=2007-12-31 | publisher=[http://pl.atyp.us/wordpress/ Canned Platypus]}}</ref> and [[fault-tolerant computing]]<ref>{{cite conference |first=Dongsheng |last=Li |coauthors=Xicheng Lu, Jinshu Su |title=Graph-Theoretic Analysis of Kautz Topology and DHT Schemes |booktitle=Network and Parallel Computing: IFIP International Conference |pages=308–315 |publisher=NPC |date=2004 |location=Wuhan, China |url=http://books.google.com/books?id=DpDwhffRCjwC&pg=PA308&lpg=PA308&dq=kautz+graph&source=web&ots=QWy7s3YiHU&sig=KHsfzBYxBWdiNjZ4pn8YUoArB0A&hl=en |accessdate=2008-03-05 | isbn=3-540-23388-1 }}</ref> applications: such a network is known as a ''Kautz network''.
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| == Notes ==
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| {{reflist}}
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| {{PlanetMath attribution|id=8526|title=Kautz graph}}
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| {{DEFAULTSORT:Kautz Graph}}
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| [[Category:Parametric families of graphs]]
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| [[Category:Directed graphs]]
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Marvella is what you can call her but it's not the most female name out there. California is our beginning place. Bookkeeping is what I do. The favorite pastime for my children and me is to perform baseball but I haven't made a dime with it.
My page ... at home std testing