Cartesian product of graphs

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The Cartesian product of graphs.

In graph theory, the Cartesian productGH of graphs G and H is a graph such that

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The operation is associative, as the graphs (FG) H and F (GH) are naturally isomorphic. The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs GH and HG 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 an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges. [1]

Examples

Thus, the Cartesian product of two hypercube graphs is another hypercube: Qi 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. [2] However, Imrich & Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:

(K1 + K2 + K22) (K1 + K23) = (K1 + K22 + K24) (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. [3]

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 H) = max {χ(G), χ(H)}. [4]

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(GH) ≤ 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(GH) ≥ γ(G)γ(H).

Cartesian product graphs can be recognized efficiently, in linear time. [5]

Algebraic graph theory

Algebraic graph theory can be used to analyse the Cartesian graph product. If the graph has vertices and the adjacency matrix , and the graph has vertices and the adjacency matrix , then the adjacency matrix of the Cartesian product of both graphs is given by

,

where denotes the Kronecker product of matrices and denotes the identity matrix. [6] The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors.

History

According to Imrich & Klavžar (2000), Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by GertSabidussi  ( 1960 ).

Notes

  1. Hahn & Sabidussi (1997).
  2. Sabidussi (1960); Vizing (1963).
  3. Imrich & Klavžar (2000), Theorem 4.19.
  4. Sabidussi (1957).
  5. Imrich & Peterin (2007). For earlier polynomial time algorithms see Feigenbaum, Hershberger & Schäffer (1985) and Aurenhammer, Hagauer & Imrich (1992).
  6. Kaveh & Rahami (2005).

References