Cartesian product of graphs

The Cartesian product of graphs.

In graph theory, the Cartesian product G \square H of graphs G and H is a graph such that

Cartesian product graphs can be recognized efficiently, in time O(m log n) for a graph with m edges and n vertices (Aurenhammer, Hagauer & Imrich 1992). The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs G \square H and H \square G are naturally isomorphic, but it is not commutative as an operation on labeled graphs. The operation is also associative, as the graphs (F \square G) \square H and F \square (G \square H) are naturally isomorphic.

The notation G × H is occasionally also used for Cartesian products of graphs, but is more commonly used for another construction known as the tensor product of graphs. The square symbol is the more common and unambiguous notation for the Cartesian product of graphs. It shows visually the four edges resulting from the Cartesian product of two edges.[1]

The Cartesian product is not a product in the category of graphs. (The tensor product is the categorical product.) However, it is a product in the category of reflexive graphs. The category of graphs does form a monoidal category under the Cartesian product.

Examples

(K_2)^{\square n} = Q_n.
Thus, the Cartesian product of two hypercube graphs is another hypercube: Qi \square Qj = Qi+j.

Properties

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 (Sabidussi 1960; Vizing 1963). However, Imrich and Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:

(K1 + K2 + K22) \square (K1 + K23) = (K1 + K22 + K24) \square (K1 + K2),

where the plus sign denotes disjoint union and the superscripts denote exponentiation over Cartesian products.

A Cartesian product is vertex transitive if and only if each of its factors is (Imrich and Klavžar, Theorem 4.19).

A Cartesian product is bipartite if and only if each of its factors is. More generally, the chromatic number of the Cartesian product satisfies the equation

χ(G \square H) = max {χ(G), χ(H)}

(Sabidussi 1957). The Hedetniemi conjecture states a related equality for the tensor product of graphs. The independence number of a Cartesian product is not so easily calculated, but as Vizing (1963) showed it satisfies the inequalities

α(G)α(H) + min{|V(G)|-α(G),|V(H)|-α(H)} ≤ α(G \square H) ≤ min{α(G) |V(H)|, α(H) |V(G)|}.

The Vizing conjecture states that the domination number of a Cartesian product satisfies the inequality

γ(G \square H) ≥ γ(G)γ(H).

Algebraic graph theory

Algebraic graph theory can be used to analyse the Cartesian graph product. If the graph G_1 has n_1 vertices and the n_1\times n_1 adjacency matrix \mathbf A_1, and the graph G_2 has n_2 vertices and the n_2\times n_2 adjacency matrix \mathbf A_2, then the adjacency matrix of the Cartesian product of both graphs is given by

\mathbf A_{1\square 2} = \mathbf A_1 \otimes \mathbf I_{n_2} + \mathbf I_{n_1} \otimes \mathbf A_2,

where \otimes denotes the Kronecker product of matrices and \mathbf I_n denotes the n\times n identity matrix.[2]

History

According to Klavžar and Imrich, Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by Sabidussi in 1960.

Notes

  1. ↑ Hahn, Geňa; Sabidussi, Gert (1997), Graph symmetry: algebraic methods and applications, NATO Advanced Science Institutes Series 497, Springer, p. 116, ISBN 978-0-7923-4668-5.
  2. ↑ Kaveh, A.; Rahami, H. (2005), "A unified method for eigendecomposition of graph products", Communications in Numerical Methods in Engineering with Biomedical Applications 21 (7): 377–388, doi:10.1002/cnm.753, MR 2151527.

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