Hi @anafilipa.carv,
Welcome to the forum!
You are correct that an adjacency matrix can be encoded directly into a GBS device consisting only of squeezers and interferometers. In the dense subgraph paper, this corresponds to Eq. (2). Very simply, the paper on dense subgraphs was written at a time where less was understood about GBS and therefore the exposition is not as clean as in the latest Nature paper.
If you want to understand more about the central concepts of programming GBS devices to run various tasks, I recommend that you take a look at this paper.
Please let me know if you have any further questions!
Juan Miguel