Replacement product: Difference between revisions
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In [[graph theory]], the '''replacement product''' of two graphs is a [[graph product]] that can be used to reduce the [[degree (graph theory)|degree]] of a graph while maintaining its [[connectivity (graph theory)|connectivity]].<ref>{{cite journal|last1=Hoory|first1=Shlomo|last2=Linial|first2=Nathan|last3=Wigderson|first3=Avi|title=Expander graphs and their applications|journal=Bulletin of the American Mathematical Society|date=7 August 2006|volume=43|issue= |
In [[graph theory]], the '''replacement product''' of two graphs is a [[graph product]] that can be used to reduce the [[degree (graph theory)|degree]] of a graph while maintaining its [[connectivity (graph theory)|connectivity]].<ref>{{cite journal|last1=Hoory|first1=Shlomo|last2=Linial|first2=Nathan|last3=Wigderson|first3=Avi|title=Expander graphs and their applications|journal=Bulletin of the American Mathematical Society|date=7 August 2006|volume=43|issue=4|pages=439–562|doi=10.1090/S0273-0979-06-01126-8}}</ref> |
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Suppose ''G'' is a ''d''-[[regular graph]] and ''H'' is an ''e''-regular graph with vertex set {0, …, ''d'' − 1}. Let ''R'' denote the replacement product of ''G'' and ''H''. The vertex set of ''R'' is the [[Cartesian product]] ''V''(''G'') × ''V''(''H''). For each vertex ''u'' in ''V''(''G'') and for each edge (''i'', ''j'') in ''E''(''H''), the vertex (''u'', ''i'') is adjacent to (''u'', ''j'') in ''R''. Furthermore, for each edge (''u'', ''v'') in ''E''(''G''), if ''v'' is the ''i''th neighbor of ''u'' and ''u'' is the ''j''th neighbor of ''v'', the vertex (''u'', ''i'') is adjacent to (''v'', ''j'') in ''R''. |
Suppose ''G'' is a ''d''-[[regular graph]] and ''H'' is an ''e''-regular graph with vertex set {0, …, ''d'' − 1}. Let ''R'' denote the replacement product of ''G'' and ''H''. The vertex set of ''R'' is the [[Cartesian product]] ''V''(''G'') × ''V''(''H''). For each vertex ''u'' in ''V''(''G'') and for each edge (''i'', ''j'') in ''E''(''H''), the vertex (''u'', ''i'') is adjacent to (''u'', ''j'') in ''R''. Furthermore, for each edge (''u'', ''v'') in ''E''(''G''), if ''v'' is the ''i''th neighbor of ''u'' and ''u'' is the ''j''th neighbor of ''v'', the vertex (''u'', ''i'') is adjacent to (''v'', ''j'') in ''R''. |
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Revision as of 12:50, 16 February 2019
In graph theory, the replacement product of two graphs is a graph product that can be used to reduce the degree of a graph while maintaining its connectivity.[1]
Suppose G is a d-regular graph and H is an e-regular graph with vertex set {0, …, d − 1}. Let R denote the replacement product of G and H. The vertex set of R is the Cartesian product V(G) × V(H). For each vertex u in V(G) and for each edge (i, j) in E(H), the vertex (u, i) is adjacent to (u, j) in R. Furthermore, for each edge (u, v) in E(G), if v is the ith neighbor of u and u is the jth neighbor of v, the vertex (u, i) is adjacent to (v, j) in R.
If H is an e-regular graph, then R is an (e + 1)-regular graph.
References
- ^ Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (7 August 2006). "Expander graphs and their applications". Bulletin of the American Mathematical Society. 43 (4): 439–562. doi:10.1090/S0273-0979-06-01126-8.
External links
- Trevisan, Luca (7 March 2011). "CS359G Lecture 17: The Zig-Zag Product". Retrieved 16 December 2014.
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