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Tensor product of graphs

Summary

In graph theory, the tensor product G × H of graphs G and H is a graph such that

The tensor product of graphs.

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.[1]

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] This product should not be confused with the strong product of graphs.

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 : IG and fH : IH yields a homomorphism

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

In the other direction, a homomorphism f : IG × H can be composed with the projections homomorphisms

${\displaystyle {\begin{cases}\pi _{G}:G\times H\to G\\\pi _{G}((u,u'))=u\end{cases}}\qquad \qquad {\begin{cases}\pi _{H}:G\times H\to H\\\pi _{H}((u,u'))=u'\end{cases}}}$

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. Imrich (1998) 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.[3] 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 Yaroslav Shitov (2019).

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 : G0H0 as vertices and an edge from f : G0H0 to f' : G0H0 whenever an edge {x, y} in G implies {f (x), f ' (y)} in H.[4]

Notes

1. ^
2. ^
3. ^ Imrich & Klavžar 2000, Theorem 5.29