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Butterfly graph: различия между версиями

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'''Butterfly graph''' --- граф-бабочка.  
'''Butterfly graph''' — ''[[граф-бабочка]].''


Let <math>n</math> be a positive integer. The <math>n</math>-level '''butterfly graph''' <math>{\mathcal B}(n)</math>
Let <math>n</math> be a positive integer. The <math>\,n</math>-level '''butterfly graph''' <math>{\mathcal B}(n)</math>
is a digraph whose vertices comprise the set <math>V_{n} = Z_{n} \times
is a [[digraph]] whose [[vertex|vertices]] comprise the set <math>V_{n} = Z_{n} \times
Z_{2}^{n}</math>. The subset <math>V_{n}^{q} = \{q\} \times Z_{2}^{n}</math> of <math>V_{n}</math>
Z_{2}^{n}</math>. The subset <math>V_{n}^{q} = \{q\} \times Z_{2}^{n}</math> of <math>\,V_{n}</math>
(<math>0 \leq q < n</math>) comprises the <math>q^{th}</math> level of <math>{\mathcal B}(n)</math>.
(<math>0 \leq q < n</math>) comprises the <math>\,q^{th}</math> level of <math>{\mathcal B}(n)</math>.


The arcs of <math>{\mathcal B}(n)</math> form directed butterflies (or, copies of the
The [[arc|arcs]] of <math>{\mathcal B}(n)</math> form directed butterflies (or, copies of the
directed complete bipartite graph <math>K_{2,2}</math>) between consecutive
directed complete [[bipartite graph]] <math>\,K_{2,2}</math>) between consecutive
levels of vertices, with wraparound in the sense that level <math>0</math> is
levels of vertices, with wraparound in the sense that level <math>\,0</math> is
identified with level <math>n</math>. Each butterfly connects each vertex
identified with level <math>\,n</math>. Each butterfly connects each vertex
<math>\langle q,\beta_{0}\beta_{1} \cdots\beta_{q-1} \alpha
<math>\langle q,\beta_{0}\beta_{1} \cdots\beta_{q-1} \alpha
\beta_{q+1}\cdots \beta_{n-1}\rangle</math>
\beta_{q+1}\cdots \beta_{n-1}\rangle</math>
on level <math>q</math> of <math>{\mathcal B}(n)</math> (<math>q \in Z_{n}</math>; <math>\alpha</math> and each
on level <math>\,q</math> of <math>{\mathcal B}(n)</math> (<math>q \in Z_{n}</math>; <math>\,\alpha</math> and each
<math>\beta_{i}</math> in <math>Z_{2}</math>) to both vertices
<math>\,\beta_{i}</math> in <math>\,Z_{2}</math>) to both vertices


<math>\langle q+1\pmod {n}, \beta_{0}\beta_{!} \cdots \beta_{q-1}\gamma
:::::<math>\langle q+1\pmod {n}, \beta_{0}\beta_{!} \cdots \beta_{q-1}\gamma \beta_{q+1}\cdots \beta_{n-1}\rangle</math>


\beta_{q+1}\cdots \beta_{n-1}\rangle</math>
on level <math> \,q+1\pmod {n} </math> of <math>{\mathcal B}(n)</math>, for <math> \,\gamma = 0, 1</math>.
 
==Литература==


on level <math> q+1\pmod {n} </math> of <math>{\mathcal B}(n)</math>, for <math> \gamma = 0, 1</math>.
* Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.