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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. The Cartesian product of two median graphs is another median graph. The graph of vertices and edges of an n-prism is the Cartesian product graph ...
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.
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.
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 operation, [ 2 ]
The graph Q 0 consists of a single vertex, while Q 1 is the complete graph on two vertices. Q 2 is a cycle of length 4. The graph Q 3 is the 1-skeleton of a cube and is a planar graph with eight vertices and twelve edges. The graph Q 4 is the Levi graph of the Möbius configuration. It is also the knight's graph for a toroidal chessboard.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs.This conjecture was first stated by Vadim G. Vizing (), and states that, if γ(G) denotes the minimum number of vertices in a dominating set for the graph G, then
For instance, the strong product of graphs, applied to any two non-empty graphs, produces complete subgraphs with four vertices, which are not unit distance graphs. The Cartesian products of path graphs form grid graphs of any dimension, the Cartesian products of the complete graph on two vertices are the hypercube graphs, [8] and the Cartesian ...