Many thanks to the authors for their submission. All reviewers have enjoyed this paper and support acceptance. I agree with reviewers that the fact that this only works for now on graphs of the same size is a problem that needs to be fixed. l.124: this is a transport map, not a plan. p.2 would benefit from some rewriting and structuring, maybe using paragraph headers. l.108: references to [26,27] are kept to the minimum. It might be worth spending a little more time on this approach to represent and play with graphs. l.116: it is important to explain that this is a distance between aligned graphs, i.e. distances between sets of edges that share the same vertices and not graphs in full generality. fig.2: right is a bit misleading, since S_\tau(\tilde{P}) seems very close to an assignment and not a bistochastic matrix. l.168: this is not exactly an abuse of notation, but rather a different way to define a covariance matrix. Altough P'LP was a simple operation (reordering of columns), here was is achieved is a mixture of the laplacian matrices lines/columns. Any considerations that is more precise could be useful.