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Concept
Cartesian product of graphs
Summary
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,u' ) and (v,v' ) are adjacent in G □ H if and only if either
u = v and u' is adjacent to v' in H, or
u' = v' and u is adjacent to v in G.
The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969].
The operation is associative, as the graphs (F □ G) □ H and F □ (G □ H) are naturally isomorphic.
The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs G □ H and H □ G are naturally isomorphic, but it is not commutative as an operation on labeled graphs.
The notation G × H has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is intended to be an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges.
The Cartesian product of two edges is a cycle on four vertices: K2□K2 = C4.
The Cartesian product of K2 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: Qi□Qj = Qi+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 K2□Cn.
The rook's graph is the Cartesian product of two complete graphs.
If a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs. However, describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:
where the plus sign denotes disjoint union and the superscripts denote exponentiation over Cartesian products.
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