How can I recognize aromatic rings in a connection table?
I will reiterate what I have written in a comment earlier, as of why I find this a bit problematic (especially for a cheminformatics approach).

There still is no clear cut definition of aromaticity, there even is an (at the time of writing) upcoming conference Aromaticity 2018 to get more clarity on the matter. 

There are a couple of questions on the site, where I have written an answer for which demonstrate the issue. I don't want to be shamelessly promoting my own posts, but I also don't want to repeat myself. I think one of the crucial examples is How to explain (non-/anti-) aromaticity in fulvene with the help of resonance structures?

From examples like this, it seems clear that from structure and Kekule form alone it appears impossible to determine whether a ring has aromatic character or not. Mithoron stated in the comments something quite categorical, which I find in its extend debatable; but I wouldn't want to dismiss it either:


  I wouldn't say 2H-indene is aromatic, much like 1,2-benzoquinone also isn't.


Take the first as an example, it has a high C-H acidity (higher than indene), because the resulting structure would have a fully conjugated system very closely related to naphthalene. This structure is undoubtedly more aromatic (handwavy, but if you could quantify it, it probably would), so any kind of structural diagram, yes/no-parameter would misrepresent that.


  Given a connection table for a molecule that contains any number of six-membered rings (possibly fused, spiro, etc.) and is represented in Kekule form, can one write an algorithm to determine which rings are aromatic and which are not? Atomic coordinates are available if necessary.


I'm afraid the answer to this question is no.



What you could do, and where you'll probably be right many times is using the bond lengths as an indicator. The bonds in such rings should all be very, very similar (maybe &pm;10%), an given Pekka Pyykkö's data (pdf from his homepage) fall somewhere around 140&nbsp;pm. Couple that with the ring-finder and I would assume it's going to be close many times.