Product of adjacency matrices I was wondering if there was any meaningful interpertation of the product of two $n\times n$ adjacency matrices of two distinct graphs.

The dot product of the adjacency matrix with itself is a measure of similarity between nodes. For instance take the non-symmetric directed adjacency matrix A = 
1, 0, 1, 0
0, 1, 0, 1
1, 0, 0, 0
1, 0, 1, 0

then the dot of $A^T$A (gram matrix) gives the un-normalized similarity between column i and column j which is the symmetric matrix:
3, 0, 2, 0
0, 1, 0, 1
2, 0, 2, 0
0, 1, 0, 1

This is much like the gram matrix of a linear kernel in an SVM. An alternate version of the kernel is the RBF kernel. The RBF kernel is simply a measure of similarity between two datapoints that can be looked up in the nxn matrix. Likewise, so is the linear kernel.
A Gram matrix is simply the dot of its transpose and itself.
Now say you have matrix B which is also a non-symmetric directed adjacency matrix. B =
1, 0, 1, 0
1, 0, 0, 0
1, 0, 0, 0
1, 0, 0, 1

So $A^T$B is a non-symmetric matrix:
3, 0, 1, 1
1, 0, 0, 0
2, 0, 1, 1
1, 0, 0, 0

Matrix A col i and matrix B col j are proportionately similar according to the above matrix. Thus the dot product of transpose of the first matrix to the second matrix is a measure of similarity of nodes characterized by their edges.
