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| In [[mathematics]], a '''graph product''' is a [[binary operation]] on [[graph (mathematics)|graph]]s. Specifically, it is an operation that takes two graphs ''G''<sub>1</sub> and ''G''<sub>2</sub> and produces a graph ''H'' with the following properties:
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| * The [[vertex (graph theory)|vertex set]] of ''H'' is the [[Cartesian product]] ''V''(''G''<sub>1</sub>) × ''V''(''G''<sub>2</sub>), where ''V''(''G''<sub>1</sub>) and ''V''(''G''<sub>2</sub>) are the vertex sets of ''G''<sub>1</sub> and ''G''<sub>2</sub>, respectively.
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| * Two vertices (''u''<sub>1</sub>, ''u''<sub>2</sub>) and (''v''<sub>1</sub>, ''v''<sub>2</sub>) of ''H'' are connected by an [[edge (graph theory)|edge]] if and only if the vertices ''u''<sub>1</sub>, ''u''<sub>2</sub>, ''v''<sub>1</sub>, ''v''<sub>2</sub> satisfy conditions of a certain type (see below).
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| The following table shows the most common graph products, with ∼ denoting “is connected by an edge to”, and <math>\not\sim</math> denoting non-connection:
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| {| class="wikitable" style="text-align:center"
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| ! Name
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| ! Condition for (''u''<sub>1</sub>, ''u''<sub>2</sub>) ∼ (''v''<sub>1</sub>, ''v''<sub>2</sub>).
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| ! Dimensions
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| ! Example
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| | [[Cartesian product of graphs|Cartesian product]]
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| | ( ''u''<sub>1</sub> = ''v''<sub>1</sub> and ''u''<sub>2</sub> ∼ ''v''<sub>2</sub> )<br/>or<br/>
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| ( ''u''<sub>1</sub> ∼ ''v''<sub>1</sub> and ''u''<sub>2</sub> = ''v''<sub>2</sub> )
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| | <math>G_{V_1, E_1} \square H_{V_2, E_2} \rightarrow J_{(V_1 V_2), (E_2 V_1 + E_1 V_2)}</math>
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| | [[Image:Graph-Cartesian-product.svg|200px]]
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| |-
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| | [[Tensor product of graphs|Tensor product]]<br/>(Categorical product)
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| | ''u''<sub>1</sub> ∼ ''v''<sub>1</sub> and ''u''<sub>2</sub> ∼ ''v''<sub>2</sub>
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| | <math>G_{V_1, E_1} \times H_{V_2, E_2} \rightarrow J_{(V_1 V_2), (2 E_1 E_2)}</math>
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| | [[Image:Graph-tensor-product.svg|200px]]
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| | [[Lexicographical product of graphs|Lexicographical product]]
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| | ''u''<sub>1</sub> ∼ ''v''<sub>1</sub><br/> or <br/>( ''u''<sub>1</sub> = ''v''<sub>1</sub> and ''u''<sub>2</sub> ∼ ''v''<sub>2</sub> )
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| | <math>G_{V_1, E_1} \cdot H_{V_2, E_2} \rightarrow J_{(V_1 V_2), (E_2 V_1 + E_1 V_2^2)}</math>
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| | [[Image:Graph-lexicographic-product.svg|200px]]
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| |-
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| | [[Strong product of graphs|Strong product]]<br/>(Normal product, AND product)
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| | ( ''u''<sub>1</sub> = ''v''<sub>1</sub> and ''u''<sub>2</sub> ∼ ''v''<sub>2</sub> )<br/>or<br/>( ''u''<sub>1</sub> ∼ ''v''<sub>1</sub> and ''u''<sub>2</sub> = ''v''<sub>2</sub> )<br/>or<br/>( ''u''<sub>1</sub> ∼ ''v''<sub>1</sub> and ''u''<sub>2</sub> ∼ ''v''<sub>2</sub> )
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| | <math>G_{V_1, E_1} \boxtimes H_{V_2, E_2} \rightarrow J_{(V_1 V_2), (V_1 E_2 + V_2E_1 + 2 E_1 E_2)}</math>
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| | [[Co-normal product of graphs|Co-normal product]]<br/>(disjunctive product, OR product)
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| | ''u''<sub>1</sub> ∼ ''v''<sub>1</sub><br/>or<br/>''u''<sub>2</sub> ∼ ''v''<sub>2</sub>
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| | [[Modular product of graphs|Modular product]]
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| | <math>(u_1 \sim v_1 \text{ and } u_2 \sim v_2)</math><br/>or</br/>
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| <math>(u_1 \not\sim v_1 \text{ and } u_2 \not\sim v_2)</math>
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| |-
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| | [[Rooted product of graphs|Rooted product]]
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| | see article
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| | <math>G_{V_1, E_1} \cdot H_{V_2, E_2} \rightarrow J_{(V_1 V_2), (E_2 V_1 + E_1)}</math>
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| | [[Image:Graph-rooted-product.svg|200px]]
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| |-
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| | [[Kronecker product]]
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| | see article
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| | see article
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| | see article
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| |-
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| | [[Zig-zag product]]
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| | see article
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| | see article
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| | see article
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| |}
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| In general, a graph product is determined by any condition for (''u''<sub>1</sub>, ''u''<sub>2</sub>) ∼ (''v''<sub>1</sub>, ''v''<sub>2</sub>) that can be expressed in terms of the statements ''u''<sub>1</sub> ∼ ''v''<sub>1</sub>, ''u''<sub>2</sub> ∼ ''v''<sub>2</sub>, ''u''<sub>1</sub> = ''v''<sub>1</sub>, and ''u''<sub>2</sub> = ''v''<sub>2</sub>.
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| ==See also==
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| * [[Graph operations]]
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| ==References==
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| *{{citation
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| | last1 = Imrich | first1 = Wilfried
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| | last2 = Klavžar | first2 = Sandi
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| | title = Product Graphs: Structure and Recognition
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| | publisher = Wiley
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| | year = 2000
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| | id = ISBN 0-471-37039-8}}.
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| {{combin-stub}}
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| [[Category:Graph products]]
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| [[Category:Graph operations]]
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| [[Category:Binary operations]]
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