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 tensor product is also called the direct product, Kronecker product, categorical product, cardinal product, relational product, weak direct product, or conjunction. As an operation on binary relations, the tensor product was introduced by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (#|1912). It is also equivalent to the Kronecker product of the adjacency matrices of the 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. This product should not be confused with the strong product of graphs.

Examples

 * The tensor product $G × K_{2}$ is a bipartite graph, called the bipartite double cover of $G$. The bipartite double cover of the Petersen graph is the Desargues graph: $K_{2} × G(5,2) = G(10,3)$. The bipartite double cover of a complete graph $K_{n}$ is a crown graph (a complete bipartite graph $K_{n,n}$ minus a perfect matching).
 * The tensor product of a complete graph with itself is the complement of a Rook's graph. Its vertices can be placed in an $n$-by-$n$ grid, so that each vertex is adjacent to the vertices that are not in the same row or column of the grid.

Properties
The tensor product is the category-theoretic product in the category of graphs and graph homomorphisms. That is, a homomorphism to $G × H$ corresponds to a pair of homomorphisms to $G$ and to $H$. In particular, a graph $I$ admits a homomorphism into $G × H$ if and only if it admits a homomorphism into $G$ and into $H$.

To see that, in one direction, observe that a pair of homomorphisms $fG : I → G$ and $fH : I → H$ yields a homomorphism


 * $$\begin{cases} f : I \to G \times H \\ f(v) = \left (f_G(v), f_H(v) \right ) \end{cases}$$

In the other direction, a homomorphism $f : I → G × H$ can be composed with the projections homomorphisms


 * $$\begin{cases} \pi_G : G \times H \to G \\ \pi_G((u,u')) = u \end{cases} \qquad \qquad \begin{cases} \pi_H : G \times H \to H \\ \pi_H((u,u')) = u' \end{cases}$$

to yield homomorphisms to $G$ and to $H$.

The adjacency matrix of $G × H$ is the Kronecker (tensor) product of the adjacency matrices of $G$ and $H$.

If a graph can be represented as a tensor product, then there may be multiple different representations (tensor products do not satisfy unique factorization) but each representation has the same number of irreducible factors. gives a polynomial time algorithm for recognizing tensor product graphs and finding a factorization of any such graph.

If either $G$ or $H$ is bipartite, then so is their tensor product. $G × H$ is connected if and only if both factors are connected and at least one factor is nonbipartite. In particular the bipartite double cover of $G$ is connected if and only if $G$ is connected and nonbipartite.

The Hedetniemi conjecture, which gave a formula for the chromatic number of a tensor product, was disproved by.

The tensor product of graphs equips the category of graphs and graph homomorphisms with the structure of a symmetric closed monoidal category. Let $G0$ denote the underlying set of vertices of the graph $G$. The internal hom $[G, H]$ has functions $f : G0 → H0$ as vertices and an edge from $f : G0 → H0$ to $f' : G0 → H0$ whenever an edge ${x, y}$ in $G$ implies ${f (x), f&thinsp;' (y)}$ in $H$.