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The Cartesian product of K 2 and a path graph is a ladder graph. The Cartesian product of two path graphs is a grid graph. The Cartesian product of n edges is a hypercube: =. Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j.
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: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v ′ in H, or v = v′ and u is adjacent with u ′ in G.
It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets: cartesian graph product: it is a commutative and associative operation (for unlabelled graphs), [2] lexicographic graph product (or graph composition): it is an associative (for unlabelled graphs) and non-commutative ...
The strong product of any two graphs can be constructed as the union of two other products of the same two graphs, the Cartesian product of graphs and the tensor product of graphs. An example of a strong product is the king's graph, the graph of moves of a chess king on a chessboard, which can be constructed as a strong product of path graphs ...
The notation G × H is also (and formerly normally was) used to represent another construction known as the Cartesian product of graphs, but nowadays more commonly refers to the tensor product. The cross symbol shows visually the two edges resulting from the tensor product of two edges. [ 2 ]
In the mathematical field of graph theory, the ladder graph L n is a planar, undirected graph with 2n vertices and 3n – 2 edges. [1] The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge: L n,1 = P n × P 2. [2] [3]
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.