1
$\begingroup$

I've tried to observe a statement from a scientific article on a practical example, but don't understand how are indices on Kronecker product mapped.

The statement is the following:

If G and H are graphs with adjacency matrices A(G) and A(H) respectively, then the Kronecker product kron(G,H) is defined as the graph with adjacency matrix kron(A(G), A(H)).

Edge(Xij,Xkl) is element of kron(G,H) iff (Xi, Xk) is element of G and (Xj,Xl) is element of H where Xij and Xkl are nodes in kron(G,H) and Xi, Xj, Xk and Xl are corresponding nodes in G and H.

In order to observe how the indices are mapped I tried to calculate the Kronecker product of A and B where A = [1 0 1 1; 0 1 0 1; 1 1 0 1; 1 1 1 1] and B = [1 0 1; 0 1 1; 1 1 0]

this results the Kronecker product K

K = kron(A, B) = [1 0 1 0 0 0 1 0 1 1 0 1; 
0 1 1 0 0 0 0 1 1 0 1 1; 
1 1 0 0 0 0 1 1 0 1 1 0; 
0 0 0 1 0 1 0 0 0 1 0 1; 
0 0 0 0 1 1 0 0 0 0 1 1; 
0 0 0 1 1 0 0 0 0 1 1 0; 
1 0 1 1 0 1 0 0 0 1 0 1; 
0 1 1 0 1 1 0 0 0 0 1 1; 
1 1 0 1 1 0 0 0 0 1 1 0; 
1 0 1 1 0 1 1 0 1 1 0 1; 
0 1 1 0 1 1 0 1 1 0 1 1; 
1 1 0 1 1 0 1 1 0 1 1 0]

which is 12x12 size matrix. Selecting nodes A(1, 3) = 1 and B(2, 3) = 1 I wanted to find the edge (12, 33) in matrix K but don't understand which row and column are represented with indices 12 and 33.

I would be thankful if anyone could explain how to find appropriate element in matrix K upon given indices from matrix A and B.

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.