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| In [[graph theory]], the '''strong product''' ''G'' ⊠ ''H'' of graphs ''G'' and ''H'' is a graph such that
| | Hi there, I am Yoshiko Villareal but I never really favored that title. Some time ago he selected to live in Idaho. Interviewing is what she does but quickly she'll be on her own. What he truly enjoys doing is to play handball but he is struggling to find time for it.<br><br>My web-site: [http://www.chillister.com/blogs/post/27388 http://www.chillister.com/blogs/post/27388] |
| * the vertex set of ''G'' ⊠ ''H'' is the Cartesian product ''V(G)'' × ''V(H)''; and
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| * any two distinct vertices ''(u,u')'' and ''(v,v')'' are adjacent in ''G'' × ''H'' [[if and only if]]:
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| ** {{mvar|u}} is adjacent to {{mvar|v}} and {{mvar|u'}}={{mvar|v'}}, or
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| ** {{mvar|u}}={{mvar|v}} and {{mvar|u'}} is adjacent to {{mvar|v'}}, or
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| ** {{mvar|u}} is adjacent to {{mvar|v}} and {{mvar|u'}} is adjacent to {{mvar|v'}}
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| The strong product is also called the '''normal product''' and '''AND product'''. It was first introduced by [[Gert Sabidussi|Sabidussi]] in 1960.
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| ==Shannon capacity and Lovász number==
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| The [[Shannon capacity of a graph]] is defined from the [[independence number]] of its strong products with itself, by the formula
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| :<math>
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| \Theta(G)
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| = \sup_k \sqrt[k]{\alpha(G^k)}
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| = \lim_{k \rightarrow \infty} \sqrt[k]{\alpha(G^k)},
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| </math> | |
| [[László Lovász]] showed that [[Lovász number|Lovász theta function]] is multiplicative:<ref>See Lemma 2 and Theorem 7 in {{harvtxt|Lovász|1979}}.</ref>
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| :<math> \vartheta(G \boxtimes H) = \vartheta(G) \vartheta(H). </math> | |
| He used this fact to upper bound the Shannon capacity by the Lovász number.
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| == See also ==
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| * [[Graph product]]
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| * [[Cartesian product of graphs]]
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| * [[Tensor product of graphs]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| *{{cite book
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| | author = Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F.
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| | title = Graphs and their Cartesian Products
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| | publisher = A. K. Peters
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| | year = 2008
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| | isbn = 1-56881-429-1}}
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| * {{citation | first = László | last = Lovász | authorlink = László Lovász | title = On the Shannon Capacity of a Graph | journal = IEEE Transactions on Information Theory | volume = IT-25 | issue = 1 | year = 1979}}.
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| *{{cite journal
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| | author = [[Gert Sabidussi|Sabidussi, G.]]
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| | title = Graph multiplication
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| | journal = Math. Z.
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| | volume = 72
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| | year = 1960
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| | pages = 446–457
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| | id = {{MathSciNet | id = 0209177}}
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| | doi = 10.1007/BF01162967}}
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| [[Category:Graph products]]
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Hi there, I am Yoshiko Villareal but I never really favored that title. Some time ago he selected to live in Idaho. Interviewing is what she does but quickly she'll be on her own. What he truly enjoys doing is to play handball but he is struggling to find time for it.
My web-site: http://www.chillister.com/blogs/post/27388