The Cartesian Product of Two Graphs

The Cartesian Product of Two Graphs

(Ed. Note: A pdf version of this article is attached at the end of the post for offline reading.)

Introduction and Preliminaries

Graphs are objects like any other, mathematically speaking. We can define operations on two graphs to make a new graph. We’ll focus in particular on a type of graph product- the Cartesian product, and its elegant connection with matrix operations. 

We mathematically define a graph G to be a set of vertices coupled with a set of edges that connect those vertices. We write G = (V_{G}, E_{G}). As an example, the left graph in Figure 1 has three vertices V_{G} = \{v_{1}, v_{2}, v_{3}\}, and two edges E_{G} = \{v_{1}v_{2}, v_{2}v_{3}\}. The order of G is the number of vertices, and the size of G is the number of edges. So our graph G has order n=3 and size m=2. This graph in particular has a special name, P_{3}, because it’s a special type of graph called a path that consists of 3 vertices. Two vertices that are connected by an edge are adjacent and denoted with the symbol \sim. 

Cartesian Product of Graphs

Now that we’ve dispensed with necessary terminology, we shall turn our attention to performing operations on two graphs to make a new graph. In particular, a type of graph multiplication called the Cartesian product. Suppose we have two graphs, G and H, with orders n_{G} and n_{H} respectively. Formally, we define the Cartesian product G \times H to be the graph with vertex set V_{G} \times V_{H}. Pause here to note what we mean by this. The vertex set of the graph Cartesian  is the Cartesian product of the vertex sets of the two graphs: V_{G \times H} = V_{G} \times V_{H}. This means that the Cartesian product of a graph has n_{G}n_{H} vertices. As an example, P_{3} \times P_{2} has 3\cdot 2 = 6 vertices. The vertices of G \times H are ordered pairs formed by V_{G} and V_{H}. For P_{3} \times P_{2} we have, 

 V_{P_{3} \times P_{2}} = \{(v_{1}, x_{1}),(v_{1}, x_{2}),(v_{2}, x_{1}),(v_{2}, x_{2}), (v_{3}, x_{1}),(v_{1}, x_{2}) \}

The edge set of G \times H is defined as follows: (v_{i}, x_{k}) is adjacent to (v_{j}, x_{l}) if

  1. v_{i} = v_{j} and x_{k} \sim x_{l}, or
  2. x_{k} = x_{l} and v_{i} \sim v_{j}

 

Let’s create P_{3} \times P_{2} as an example:

 

The red edges are due to condition (1) above, and the red to (2). Interestingly enough, this operation is commutative up to isomorphism (a relabeling of vertices that maintains the graph structure). This will be examined in a later section.

Graph and Matrices

We can also operate on graphs using matrices. The pictures above are one way to represent a graph. An adjacency matrix is another. The adjacency matrix A_{G} of a graph G of order n is an n \times n square matrix whose entries a_{ij} are given by 

a_{ij} = \begin{cases} 1, & v_{i} \sim v_{j} \\0, & i=j \text{ or } v_{i} \not\sim v_{j}\end{cases}.

 The adjacency matrix for P_{3} and P_{2} respectively are 

A_{P_{3}} = \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\0 & 1 & 0\end{bmatrix} \qquad A_{P_{2}} = \begin{bmatrix} 0 & 1\\1 & 0\end{bmatrix}

Note that a relabeling of vertices simply permutes the rows and columns of the adjacency matrix. 

Adjacency Matrix of the Cartesian product of Graphs

What is the adjacency matrix of G \times H? If G and H have orders n_{G} and n_{H} respectively, we can show that

A_{G \times H} = A_{G} \otimes I_{n_{H}} + I_{n_{G}} \otimes A_{H}

where \otimes is the Kronecker product, and I_{m} is the m\times m identity matrix. Recall that the Kronecker product of two matrices A_{m \times n} \otimes B_{k \times l} is the mk \times nl matrix given by 

A \otimes B = \begin{bmatrix} a_{11}B & a_{12}B &\ldots & a_{1n}B \\ a_{21}B & a_{22}B & \ldots & a_{2n}B \\ \vdots & & \ddots& \vdots \\ a_{m1}B & a_{m2}B & \ldots & a_{mn}B\end{bmatrix}

In general, the Kronecker product is not commutative, so A \otimes B \neq B \otimes A. 

We can prove the formula for A_{G \times H} above using standard matrix theory, but this really wouldn’t be that illuminating. We’d see that the statement is indeed true, but we would gain very little insight into how this affects the graph and why. We shall break apart the formula to see why the Kronecker product is needed, and what the addition is doing. 

Let us return to finding P_{3} \times P_{2}. By our formula, A_{P_{3} \times P_{2}} = A_{P_{3}} \otimes I_{2} + I_{3} \otimes A_{P_{2}}. Notice first why the dimensions of the identity matrices “cross” each other; that is, we use the identity matrix of order n_{P_{2}} in the Kronecker product with A_{P_{3}} and vice versa n_{P_{3}} for A_{P_{2}}. This ensures we get the correct dimension for the adjacency matrix of P_{3} \times P_{2}, which is 6 \times 6. 

We’ll walk through the computation one term at a time. First,

\begin{aligned}A_{P_{3}} \otimes I_{2} &= \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\0 & 1 & 0\\\end{bmatrix} \otimes \begin{bmatrix} 1 & 0\\0 & 1\end{bmatrix} \\&= \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0\\\end{bmatrix}\end{aligned}

This matrix is the adjacency matrix of a 6-vertex graph given below:

Notice that we now have two copies of P_{3} now in one graph. Similarly, I_{3} \otimes A_{P_{2}} is 

\begin{aligned}I_{3} \otimes A_{P_{2}} &= \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}\end{aligned}

 

Here the second Kronecker product term has made three copies of P_{2}. The last step is to add these two matrices together to obtain the adjacency matrix for P_{3} \times P_{2}. Graphically, what’s happening? This addition term overlays our 3 copies of P_{2} (usually written 3P_{2} or P_{2} \cup P_{2} \cup P_{2}) onto the respectively labeled vertices of our two copies of P_{3} (written 2P_{3} or P_{3} \cup P_{3}). That is,

\begin{aligned}A_{P_{3} \times P_{2}} &= A_{P_{3}} \otimes I_{2} + I_{3} \otimes A_{P_{2}}\\&= \begin{bmatrix} 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 1 & 0\end{bmatrix}\end{aligned}

which is the adjacency matrix of our originally obtained graph

 

Note that here the vertex labels aren’t ordered pairs. That’s ok. Technically we shouldn’t have labeled the vertices of the two graphs identically, but the goal was to illustrate the action of the Cartesian product. We can always relabel the edges. Formally, the overlaying of 3P_{2} onto the vertices of 2P_{3} would be forming those ordered-pair vertices. 

Commutativity of the Graph Cartesian Product

A natural question for any operation is whether or not it possesses the property of commutativity. An operation is commutative if the order in which we take it produces the same result. That is, if \square is an operation, a\square b = b\square a.

Does G \times H yield the same graph as H \times G? In a way, yes. Let’s examine what happens if we switch the order of the graph Cartesian product and take P_{2} \times P_{3}. 

We won’t go quite as thoroughly through each step, and instead give the results. 

We know the adjacency matrix of P_{2} \times P_{3} is given by 

A_{P_{2} \times P_{3}} = A_{P_{2}} \otimes I_{3} + I_{2} \otimes A_{P_{3}} \begin{aligned}A_{P_{2}} \otimes I_{3} &= \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0\end{bmatrix}\end{aligned}

Notice again we have 3 copies of P_{2}, but the labeling and drawing is now different. If we create a mapping \phi that relabels v_{4} as v_{2}, v_{6} as v_{4}, and v_{2} as v_{6}, then we’d get back the graph we drew from the adjacency matrix formed by I_{3} \otimes A_{P_{2}}. Thus, the two graphs are structurally equivalent; it just looks different due to labeling and redrawing. The mapping \phi is called an isomorphism because it transforms one graph into the other without losing any structural properties, and the two graphs are isomorphic. 

This implies that A_{P_{2}} \otimes I_{3} \neq I_{3} \otimes A_{P_{2}}, but if we permute rows and columns, we can transform one matrix into the other. If the labelings on the vertices didn’t matter, then the operation is commutative up to isomorphism. If the labelings matter (more common in engineering), then we do not get the “same” graph by switching the order of the Kronecker product.

We’ll do the same thing and examine I_{2} \otimes A_{P_{3}} and compare it to A_{P_{3}}\otimes I_{2}:

\begin{aligned}I_{2} \otimes A_{P_{3}} &= \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0\\1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}\end{aligned}

The corresponding graph is 

Notice again that here we have two copies of P_{3} just as when we created the graph formed by the matrix A_{P_{3}}\otimes I_{2}, but the labels are again permuted. We have generated an isomorphic graph. 

Finally, we add the two matrices together (or follow the definition to create the last step of the Cartesian product by overlaying vertices and fusing them together). 

\begin{aligned}A_{P_{2}} \otimes I_{3} + I_{2} \otimes A_{P_{3}} &= \begin{bmatrix}0 & 1 & 0 & 1 & 0 & 0\\1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \end{bmatrix}\end{aligned}

This is isomorphic to P_{3} \times P_{2}. We can show this by finding a mapping \phi that “untangles” P_{2} \times P_{3} by relabeling vertices. We can also redraw the ladder structure we saw with P_{3} \times P_{2} and label the vertices so that we get the structure of P_{2} \times P_{3}

Our conclusion is that the graph Cartesian product is “sort of” commutative as an operation. If the vertices were unlabeled or didn’t matter, then the operation is commutative because we can always relabel vertices or “untangle” the structure. If the vertices are labeled and matter, then we don’t get the same graph back when we switch the order in the Cartesian product. Thus, we can say that the graph Cartesian product is commutative to isomorphism. 

Continuation

A natural question after looking through all of this is the following:

Given an adjacency matrix, can we decompose it into a Cartesian product of two graphs?

The next article will address this.

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