TITLE: Orbits of adjacency matrices under conjugation by permutation matrices.
QUESTION [5 upvotes]: (Disclaimer: I am new here, so be patient with my mistakes, but I welcome corrections, advice or comments.)
I am interested in if anyone knows of ways of characterizing the orbits of an adjacency matrix of a multi-graph when it is conjugated by a permutation matrix. To clarify: The matrices in question are symmetric, with zero diagonal (sometimes called hollow symmetric), and all entries non-negative integers. 
Conjugation by a permutation matrix will give another adjacency matrix with all the same properties. From a certain perspective this can be visualized as re-drawing the graph that gave rise to the adjacency matrix (in a manner that is isomorphic in the graph-theory sense).
I want to know if, given two adjacency matrices with the same dimensions, is it possible to determine if they lie in the same orbit (without re-constructing the necessary permutation matrices). However, I am interested in any results that are related to this question, since I believe the answer is not known (correct me if I am wrong here).

REPLY [4 votes]: You are asking for an algorithm to determine if two multigraphs given by adjacency matrices are equivalent.
The problem of graph isomorphism is a notorious one in complexity theory, since it is only known to sit somewhere inbetween P and NP, but it is not known if it is P or NP.
From this it is clear that the problem is not an easy one.
The best existing program for graph isomorphism problems is nauty. However, I don't know if it supports multigraphs out of the box. There are interfaces to nauty in Sage and Magma.