Complete graph invariant based on integer programming?

Question

Roughly speaking, we are trying to find complete graph invariant as the lexicographically first matrix from the permutations of the adjacency matrix.

Let $G$ be graph, possibly directed graph, of order $n$ and adjacency matrix $A_G$.

Let $P$ be matrix with entries variables $x_{i,j}$. Add the linear integer constraints $P$ to be permutation matrix: $x_{i,j}$ are non-negative and each row and each column sum to $1$.

Define $f(i,j)=2^{i n + j}$.

Let $B=P A_G$. The entries of $B$ are linear equations in the variables of $P$.

Take the optimization problem:

$F(G) =\text{ minimize } \sum_{0 \le i,j \le n-1} f(i,j) B[i,j]$.

Is the above integer linear program complete graph invariant, i.e. $F(G)=F(H)$ iff $G,H$ are isomorphic?

For all subsets $i,j$ with $B[i,j]=1$ the sums $f(i,j)B[i,j]$ are distinct.

We believe the RHS of $F(G)$ is bijection matrix with 0-1 entries and the integers $[0,2^{n^2-1}]$.

In case of negative answer can we take $B=P A_G P^{-1}=P A_G P^T$ and get quadratic integer program?

Added

We did experiments on small graphs. The approach $B=P A_G$ doesn't work, but the approach $B=P A_G P^T$ works on the tested graphs.

Since we don't have solver for optimizing quadratic function, we enumerated the permutations.

Here is sage code that can be run in a browser.

def mafai(g):
    n=g.order()
    mi=oo

    A=g.adjacency_matrix()

    for pe in Permutations(n):
        P=pe.to_matrix()
        B=P*A*P.transpose()
        su=0
        for i in xrange(n):
            for j in xrange(n):
                su += B[i,j]*2**(i*n+j)
        mi=min(su,mi)       
    return mi