Returns the adjacency matrix of the cartesian product of two graphs given the adjacency matrix of each one, \(G\) and \(H\).
Arguments
- G
adjacency matrix of the first graph.
- H
adjacency matrix of the second graph. If not provided, it takes the same value as
G.
Value
Let \(A(G),\ A(H)\) be the adjacency matrices of the graphs \(G,\ H\) such that \(|V(G)| = n\) and \(|V(H)| = m\), then the adjacency matrix of the cartesian product \(G \times H\) is given by
$$A(G \times H) = A(G) \otimes I_{m\ x\ m} + I_{n\ x\ n} \otimes A(H)$$
Examples
P3 <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
K3 <- matrix(c(0,1,1,1,0,1,1,1,0), nrow=3)
# Return the adjacency matrix of P3 X K3
cartesian(P3, K3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
#> [1,] 0 1 1 1 0 0 0 0 0
#> [2,] 1 0 1 0 1 0 0 0 0
#> [3,] 1 1 0 0 0 1 0 0 0
#> [4,] 1 0 0 0 1 1 1 0 0
#> [5,] 0 1 0 1 0 1 0 1 0
#> [6,] 0 0 1 1 1 0 0 0 1
#> [7,] 0 0 0 1 0 0 0 1 1
#> [8,] 0 0 0 0 1 0 1 0 1
#> [9,] 0 0 0 0 0 1 1 1 0
# Return the adjacency matrix of P3 X P3
cartesian(P3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
#> [1,] 0 1 0 1 0 0 0 0 0
#> [2,] 1 0 1 0 1 0 0 0 0
#> [3,] 0 1 0 0 0 1 0 0 0
#> [4,] 1 0 0 0 1 0 1 0 0
#> [5,] 0 1 0 1 0 1 0 1 0
#> [6,] 0 0 1 0 1 0 0 0 1
#> [7,] 0 0 0 1 0 0 0 1 0
#> [8,] 0 0 0 0 1 0 1 0 1
#> [9,] 0 0 0 0 0 1 0 1 0