• The tensor product G × K₂ is a bipartite graph, called the bipartite double cover of G. The bipartite double cover of the Petersen graph is the Desargues graph: K₂ × 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.

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 f_G : I → G and f_H : I → H 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 : I → G × 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 : 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 ' (y)} in H.^[4]

• Graph product
• Strong product of graphs

• Brown, R.; Morris, I.; Shrimpton, J.; Wensley, C. D. (2008), "Graphs of Morphisms of Graphs", The Electronic Journal of Combinatorics, 15: A1.
• Hahn, Geňa; Sabidussi, Gert (1997), Graph symmetry: algebraic methods and applications, NATO Advanced Science Institutes Series, vol. 497, Springer, p. 116, ISBN 978-0-7923-4668-5.
• Imrich, W. (1998), "Factoring cardinal product graphs in polynomial time", Discrete Mathematics, 192: 119–144, doi:10.1016/S0012-365X(98)00069-7, MR 1656730
• Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN 0-471-37039-8
• Shitov, Yaroslav (May 2019), Counterexamples to Hedetniemi's conjecture, arXiv:1905.02167
• Weichsel, Paul M. (1962), "The Kronecker product of graphs", Proceedings of the American Mathematical Society, 13 (1): 47–52, doi:10.2307/2033769, JSTOR 2033769, MR 0133816
• Whitehead, A. N.; Russell, B. (1912), Principia Mathematica, Cambridge University Press, vol. 2, p. 384

• Nicolas Bray. "Graph Categorical Product". MathWorld.