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A Cartesian product of two graphs. In graph theory, the Cartesian product G H of graphs G and H is a graph such that: the vertex set of G H is the Cartesian product V(G) × V(H); and; two vertices (u,v) and (u' ,v' ) are adjacent in G H if and only if either u = u' and v is adjacent to v' in H, or; v = v' and u is adjacent to 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.
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 tensor product of graphs. In graph theory, the tensor product G × H of graphs G and H is a graph such that the vertex set of G × H is the Cartesian product V(G) × V(H); and; vertices (g,h) and (g',h' ) are adjacent in G × H if and only if. g is adjacent to g' in G, and; h is adjacent to h' in H.
The lexicographic product of graphs. In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the cartesian product V(G) × V(H); and; any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is ...
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 ...
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