2001:4CA0:4103:F:CABC:C8FF:FE97:DCFE at 16:33, 23 September 2014

2014-09-23T16:33:48Z

en>Testem: rm disambiguation link

2014-03-04T10:09:30Z

rm disambiguation link

← Older revision		Revision as of 12:09, 4 March 2014
Line 1:		Line 1:
	[~~[Image~~:~~Graph-rooted~~-~~product.svg\|thumb\|300px\|The rooted product of graphs~~.]]		Bryan is actually a celebrity during the building and the profession expansion 1st 2nd to his 3rd place album, & , will be the resistant. He burst open to the picture in 2005 with his crazy mix of down-home availability, [http://lukebryantickets.sgs-suparco.org luke bryan concert tickets 2014] motion picture star good appearance and lines, is scheduled t within a significant way. The new recording Top about the country graph or chart and #2 about the take graphs, earning it the next highest debut during that time of 2003 for a region designer. <br><br>The boy of any , knows perseverance and determination are important elements in terms of an effective occupation- . His to start with album, Keep [http://www.cinemaudiosociety.org where can i buy luke bryan tickets] Me, generated the Top strikes “All My Friends Say” and “Country Gentleman,” although his energy, Doin’ Factor, found the performer-about three right No. 1 single people: In addition Calling Is usually a Very good Factor.”<br><br>In the fall of 2012, Concert tours: Luke Bryan And which in fact had an outstanding selection of , which include Urban. “It’s almost like you’re getting a acceptance to travel to a higher level, claims those designers that have been a part of the Tourover right into a bigger degree of performers.” It packaged as the best excursions in their 15-season historical past.<br><br>my site - [http://www.netpaw.org luke bryan tour schedule 2014]
	~~In mathematical [[graph theory]]~~, the ~~'''rooted product''' of a [[~~graph ~~(mathematics)\|graph]] ''G''~~ and ~~a [[rooted graph]] ''H'' is defined as follows:~~ take ~~\|''V''(''G'')\| copies~~ of ~~''H'', and~~ for ~~every vertex~~ <~~math~~>~~v_i~~<~~/math~~> of ~~''G''~~, ~~identify <math>v_i</math> with the root node~~ of ~~the ''i''~~-~~th copy of ''H''~~.

	~~More formally~~, ~~assuming that ''V''(''G'') = {''g''<sub>1</sub>, ..., ''g''<sub>''n''<~~/~~sub>}, ''V''(''H'') = {''h''<sub>1<~~/~~sub>,~~ ..~~., ''h''<sub>''m''</sub>} and that the root node of ''H'' is <math>h_1</math>, define~~

	~~:<math>G \circ H := (V, E)</math>~~

	where

	~~:<math>V = \left\{(g_i, h_j): 1\leq~~ i~~\leq n~~, ~~1\leq j\leq m\right\}</math>~~

	and

	~~:<math>E = \left\{((g_i~~, ~~h_1)~~, ~~(g_k~~, ~~h_1)): (g_i, g_k) \in E(G)\~~right~~\} \cup \bigcup_{i=~~1~~}^n \left\{((g_i, h_j), (g_i, h_k))~~: ~~(h_j, h_k) \in E(H)\right\}~~<~~/math~~>

	~~If ''G'' is also rooted at ''g''~~<~~sub~~>~~1</sub>, one can view~~ the ~~product itself as rooted~~, ~~at (''g''<sub>1</sub>~~, ~~''h''<sub>1</sub>)~~. ~~The rooted product is~~ a ~~[[Glossary of graph theory#Subgraphs\|subgraph]]~~ of the ~~[[cartesian product~~ of ~~graphs\|cartesian product]] of the same two graphs~~.

	~~==Applications==~~
	~~The rooted product is especially relevant for [[tree (graph theory)\|trees]],~~ as the ~~rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find [[Edge~~-~~graceful labeling\|graceful numberings]] for a wide family of trees~~.

	~~If ''H'' is a two-vertex complete graph ''K''~~<~~sub~~>2<~~/sub~~>, then for any graph ''G'', the rooted product of ''G'' and ''H'' has [[dominating set\|domination number]] exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-~~vertex [~~[cycle graph]]. These graphs can be used to generate examples in which the bound of [[Vizing's conjecture]], an unproven inequality between the domination number of the graphs in a different graph product, the [[cartesian product of graphs]], is exactly met {{harv\|Fink\|Jacobson\|Kinch\|Roberts\|1985}}. They are also [[well-covered graph]]s.

	~~==References==~~
	*{{citation
	~~\| last1 = Godsil \| first1 = C. D. \| author1-link = Chris Godsil \| author2-link = Brendan McKay \| last2 = McKay \| first2 = B. D.~~
	~~\| title = A new graph product and its spectrum~~
	~~\| journal = Bull. Austral. Math. Soc.~~
	~~\| volume = 18~~
	~~\| year = 1978~~
	~~\| issue = 1~~
	~~\| pages = 21–28~~
	~~\| mr = 0494910~~
	~~\| url =~~ http://~~cs.anu~~.~~edu~~.~~au/~bdm/papers/RootedProduct.pdf~~
	~~\| doi = 10.1017/S0004972700007760}}.~~
	*{{citation
	~~\| last1 = Fink \| first1 = J. F.~~
	~~\| last2 = Jacobson \| first2 = M. S.~~
	~~\| last3 = Kinch \| first3 = L. F.~~
	~~\| last4 = Roberts \| first4 = J.~~
	~~\| title = On graphs having domination number half their order~~
	~~\| year = 1985~~
	~~\| journal = Period. Math. Hungar.~~
	~~\| volume = 16~~
	~~\| pages = 287–293~~
	~~\| mr = 0833264~~
	~~\| doi = 10.1007/BF01848079~~
	~~\| issue = 4}}.~~
	*{{citation
	~~\| last1 = Koh \| first1 = K. M. \| last2 = Rogers \| first2 = D. G. \| last3 = Tan \| first3 = T.~~
	~~\| title = Products of graceful trees~~
	~~\| journal = [[Discrete Mathematics (journal)\|Discrete Mathematics]]~~
	~~\| volume = 31~~
	~~\| year = 1980~~
	~~\| issue = 3~~
	~~\| pages = 279–292~~
	~~\| mr = 0584121~~
	~~\| doi = 10.1016/0012-365X(80)90139-9}}.~~

	~~[[Category:Graph products]~~]

en>Mark viking: /* In and Out fields */ Added wl, spelling fix

2013-06-18T03:15:18Z

In and Out fields: Added wl, spelling fix

Show changes

en>Jan Krieg: Removed vandalism

2011-12-05T12:58:01Z

Removed vandalism

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← Older revision		Revision as of 12:09, 4 March 2014
Line 1:		Line 1:
	[~~[Image~~:~~Graph-rooted~~-~~product.svg\|thumb\|300px\|The rooted product of graphs~~.]]		Bryan is actually a celebrity during the building and the profession expansion 1st 2nd to his 3rd place album, & , will be the resistant. He burst open to the picture in 2005 with his crazy mix of down-home availability, [http://lukebryantickets.sgs-suparco.org luke bryan concert tickets 2014] motion picture star good appearance and lines, is scheduled t within a significant way. The new recording Top about the country graph or chart and #2 about the take graphs, earning it the next highest debut during that time of 2003 for a region designer. <br><br>The boy of any , knows perseverance and determination are important elements in terms of an effective occupation- . His to start with album, Keep [http://www.cinemaudiosociety.org where can i buy luke bryan tickets] Me, generated the Top strikes “All My Friends Say” and “Country Gentleman,” although his energy, Doin’ Factor, found the performer-about three right No. 1 single people: In addition Calling Is usually a Very good Factor.”<br><br>In the fall of 2012, Concert tours: Luke Bryan And which in fact had an outstanding selection of , which include Urban. “It’s almost like you’re getting a acceptance to travel to a higher level, claims those designers that have been a part of the Tourover right into a bigger degree of performers.” It packaged as the best excursions in their 15-season historical past.<br><br>my site - [http://www.netpaw.org luke bryan tour schedule 2014]
	~~In mathematical [[graph theory]]~~, the ~~'''rooted product''' of a [[~~graph ~~(mathematics)\|graph]] ''G''~~ and ~~a [[rooted graph]] ''H'' is defined as follows:~~ take ~~\|''V''(''G'')\| copies~~ of ~~''H'', and~~ for ~~every vertex~~ <~~math~~>~~v_i~~<~~/math~~> of ~~''G''~~, ~~identify <math>v_i</math> with the root node~~ of ~~the ''i''~~-~~th copy of ''H''~~.

	~~More formally~~, ~~assuming that ''V''(''G'') = {''g''<sub>1</sub>, ..., ''g''<sub>''n''<~~/~~sub>}, ''V''(''H'') = {''h''<sub>1<~~/~~sub>,~~ ..~~., ''h''<sub>''m''</sub>} and that the root node of ''H'' is <math>h_1</math>, define~~

	~~:<math>G \circ H := (V, E)</math>~~

	where

	~~:<math>V = \left\{(g_i, h_j): 1\leq~~ i~~\leq n~~, ~~1\leq j\leq m\right\}</math>~~

	and

	~~:<math>E = \left\{((g_i~~, ~~h_1)~~, ~~(g_k~~, ~~h_1)): (g_i, g_k) \in E(G)\~~right~~\} \cup \bigcup_{i=~~1~~}^n \left\{((g_i, h_j), (g_i, h_k))~~: ~~(h_j, h_k) \in E(H)\right\}~~<~~/math~~>

	~~If ''G'' is also rooted at ''g''~~<~~sub~~>~~1</sub>, one can view~~ the ~~product itself as rooted~~, ~~at (''g''<sub>1</sub>~~, ~~''h''<sub>1</sub>)~~. ~~The rooted product is~~ a ~~[[Glossary of graph theory#Subgraphs\|subgraph]]~~ of the ~~[[cartesian product~~ of ~~graphs\|cartesian product]] of the same two graphs~~.

	~~==Applications==~~
	~~The rooted product is especially relevant for [[tree (graph theory)\|trees]],~~ as the ~~rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find [[Edge~~-~~graceful labeling\|graceful numberings]] for a wide family of trees~~.

	~~If ''H'' is a two-vertex complete graph ''K''~~<~~sub~~>2<~~/sub~~>, then for any graph ''G'', the rooted product of ''G'' and ''H'' has [[dominating set\|domination number]] exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-~~vertex [~~[cycle graph]]. These graphs can be used to generate examples in which the bound of [[Vizing's conjecture]], an unproven inequality between the domination number of the graphs in a different graph product, the [[cartesian product of graphs]], is exactly met {{harv\|Fink\|Jacobson\|Kinch\|Roberts\|1985}}. They are also [[well-covered graph]]s.

	~~==References==~~
	*{{citation
	~~\| last1 = Godsil \| first1 = C. D. \| author1-link = Chris Godsil \| author2-link = Brendan McKay \| last2 = McKay \| first2 = B. D.~~
	~~\| title = A new graph product and its spectrum~~
	~~\| journal = Bull. Austral. Math. Soc.~~
	~~\| volume = 18~~
	~~\| year = 1978~~
	~~\| issue = 1~~
	~~\| pages = 21–28~~
	~~\| mr = 0494910~~
	~~\| url =~~ http://~~cs.anu~~.~~edu~~.~~au/~bdm/papers/RootedProduct.pdf~~
	~~\| doi = 10.1017/S0004972700007760}}.~~
	*{{citation
	~~\| last1 = Fink \| first1 = J. F.~~
	~~\| last2 = Jacobson \| first2 = M. S.~~
	~~\| last3 = Kinch \| first3 = L. F.~~
	~~\| last4 = Roberts \| first4 = J.~~
	~~\| title = On graphs having domination number half their order~~
	~~\| year = 1985~~
	~~\| journal = Period. Math. Hungar.~~
	~~\| volume = 16~~
	~~\| pages = 287–293~~
	~~\| mr = 0833264~~
	~~\| doi = 10.1007/BF01848079~~
	~~\| issue = 4}}.~~
	*{{citation
	~~\| last1 = Koh \| first1 = K. M. \| last2 = Rogers \| first2 = D. G. \| last3 = Tan \| first3 = T.~~
	~~\| title = Products of graceful trees~~
	~~\| journal = [[Discrete Mathematics (journal)\|Discrete Mathematics]]~~
	~~\| volume = 31~~
	~~\| year = 1980~~
	~~\| issue = 3~~
	~~\| pages = 279–292~~
	~~\| mr = 0584121~~
	~~\| doi = 10.1016/0012-365X(80)90139-9}}.~~

	~~[[Category:Graph products]~~]

LSZ reduction formula - Revision history

2001:4CA0:4103:F:CABC:C8FF:FE97:DCFE at 16:33, 23 September 2014

en>Testem: rm disambiguation link

en>Mark viking: /* In and Out fields */ Added wl, spelling fix

en>Jan Krieg: Removed vandalism