In graph theory , a graph product is a binary operation on graphs . Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties:
16-838: [REDACTED] Look up product in Wiktionary, the free dictionary. Product may refer to: Business [ edit ] Product (business) , an item that can be offered to a market to satisfy the desire or need of a customer. Product (project management) , a deliverable or set of deliverables that contribute to a business solution Mathematics [ edit ] Product (mathematics) Algebra [ edit ] Direct product Set theory [ edit ] Cartesian product of sets Group theory [ edit ] Direct product of groups Semidirect product Product of group subsets Wreath product Free product Zappa–Szép product (or knit product),
32-447: A monoidal category Product (category theory) , a generalization of mathematical products Fibre product or pullback Coproduct or pushout Probability theory [ edit ] Wick product of random variables Graph theory [ edit ] Graph product Music [ edit ] Product (Brand X album) , 1979 Product (De Press album) , 1982 Product (Sophie album) , 2015 Product ,
48-447: A monoidal category Product (category theory) , a generalization of mathematical products Fibre product or pullback Coproduct or pushout Probability theory [ edit ] Wick product of random variables Graph theory [ edit ] Graph product Music [ edit ] Product (Brand X album) , 1979 Product (De Press album) , 1982 Product (Sophie album) , 2015 Product ,
64-811: A generalization of the direct and semidirect products Ring theory [ edit ] Product of rings Ideal operations, for product of ideals Linear algebra [ edit ] Scalar multiplication Matrix multiplication Inner product, on an inner product space Exterior product or wedge product Multiplication of vectors : Dot product Cross product Seven-dimensional cross product Triple product , in vector calculus Tensor product Topology [ edit ] Product topology Algebraic topology [ edit ] Cap product Cup product Slant product Homotopy theory [ edit ] Smash product Wedge sum (or wedge product) Category theory [ edit ] Internal product, in
80-811: A generalization of the direct and semidirect products Ring theory [ edit ] Product of rings Ideal operations, for product of ideals Linear algebra [ edit ] Scalar multiplication Matrix multiplication Inner product, on an inner product space Exterior product or wedge product Multiplication of vectors : Dot product Cross product Seven-dimensional cross product Triple product , in vector calculus Tensor product Topology [ edit ] Product topology Algebraic topology [ edit ] Cap product Cup product Slant product Homotopy theory [ edit ] Smash product Wedge sum (or wedge product) Category theory [ edit ] Internal product, in
96-543: A graph product is determined by any condition for ( a 1 , a 2 ) ∼ ( b 1 , b 2 ) {\displaystyle (a_{1},a_{2})\sim (b_{1},b_{2})} that can be expressed in terms of a n = b n {\displaystyle a_{n}=b_{n}} and a n ∼ b n {\displaystyle a_{n}\sim b_{n}} . Let K 2 {\displaystyle K_{2}} be
112-424: A three-CD compilation set by Buzzcocks Other uses [ edit ] Product (chemistry) , the species formed from chemical reactions Hairstyling product , such as hairspray or hair gel, generally referred to as simply "product" See also [ edit ] Production (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
128-424: A three-CD compilation set by Buzzcocks Other uses [ edit ] Product (chemistry) , the species formed from chemical reactions Hairstyling product , such as hairspray or hair gel, generally referred to as simply "product" See also [ edit ] Production (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
144-430: Is always about whether or not the vertices a n , b n in G n are equal or connected by an edge. The terminology and notation for specific graph products in the literature varies quite a lot; even if the following may be considered somewhat standard, readers are advised to check what definition a particular author uses for a graph product, especially in older texts. Even for more standard definitions, it
160-408: Is not always consistent in the literature how to handle self-loops . The formulas below for the number of edges in a product also may fail when including self-loops. For example, the tensor product of a single vertex self-loop with itself is another single vertex self-loop with E = 1 {\displaystyle E=1} , and not E = 2 {\displaystyle E=2} as
176-422: The complete graph on two vertices (i.e. a single edge). The product graphs K 2 ◻ K 2 {\displaystyle K_{2}\square K_{2}} , K 2 × K 2 {\displaystyle K_{2}\times K_{2}} , and K 2 ⊠ K 2 {\displaystyle K_{2}\boxtimes K_{2}} look exactly like
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#1732773316504192-727: The formula E G × H = 2 E G E H {\displaystyle E_{G\times H}=2E_{G}E_{H}} would suggest. The following table shows the most common graph products, with ∼ {\displaystyle \sim } denoting "is connected by an edge to", and ≁ {\displaystyle \not \sim } denoting non-adjacency. While ≁ {\displaystyle \not \sim } does allow equality, ≄ {\displaystyle \not \simeq } means they must be distinct and non-adjacent. The operator symbols listed here are by no means standard, especially in older papers. In general,
208-763: The 💕 [REDACTED] Look up product in Wiktionary, the free dictionary. Product may refer to: Business [ edit ] Product (business) , an item that can be offered to a market to satisfy the desire or need of a customer. Product (project management) , a deliverable or set of deliverables that contribute to a business solution Mathematics [ edit ] Product (mathematics) Algebra [ edit ] Direct product Set theory [ edit ] Cartesian product of sets Group theory [ edit ] Direct product of groups Semidirect product Product of group subsets Wreath product Free product Zappa–Szép product (or knit product),
224-486: The graph representing the operator. For example, K 2 ◻ K 2 {\displaystyle K_{2}\square K_{2}} is a four cycle (a square) and K 2 ⊠ K 2 {\displaystyle K_{2}\boxtimes K_{2}} is the complete graph on four vertices. The G 1 [ G 2 ] {\displaystyle G_{1}[G_{2}]} notation for lexicographic product serves as
240-455: The title Product . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Product&oldid=1233931032 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages product From Misplaced Pages,
256-508: The title Product . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Product&oldid=1233931032 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Graph product The graph products differ in what exactly this condition is. It
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