An analogue to OEIS for graphs and ordered pairs? Does anyone know if such a thing exists already? I was able to find a number of seemingly very comprehensive encyclopedia-like reference pages on many species of graphs, but it seemed that the use case and spirit behind these pages differed from the the more Wiki-like OEIS.
If such a resource exists, please point me at it. Ideally, a database that is searchable by graph or subgraph along the same lines as OEIS with its subsequences (a simple list of ordered pairs should suffice for many queries), and then allows for accumulated community comments for that graph, though I realize this differs in that it may only be computationally tractable for relatively small structures.
If it does not, I am seriously considering trying to establish one myself, iff this is the sort of thing that people like you would be interested in using. I realize this is a questionable post for this forum; that said, I am trying to gauge the interest in a potentially useful free mathematical public good, which seems like a reasonable cause to post, and I can think of absolutely no more relevant target audience to query on this. Even the probable need to handle some method of graph canonization seems involved at the very least. I'm still unsure as to whether this is a great or terrible idea (or if there is interest), so a little bit of feedback either way would be tremendously helpful.
 A: The House of Graphs is the analogue you're looking for, at least in spirit.
It does not allow searches by subgraphs, but it does allow searches by various graph properties - in my experience, if you have a graph, and you want to see if it has an entry in the House of Graphs database, it is pretty straightforward to narrow things down enough. (You can also draw the graph or paste a graph6 string, if you're looking for a specific graph.)
More usefully, it can often tell you if a graph exists with the properties you want. For example, this question, this question, and this question are all MSE questions I answered just by searching the House of Graphs database.
Anyone can add graphs or comments to them after creating an account, though currently not many of the graphs have detailed comments.
A: Second the reference to House of Graphs, as that's the closest that I know of for what you are looking for.
However, the analogous service would be searchable chemical databases. For example, chemspider:
http://www.chemspider.com/
which allows for exact structure searches, as well as substructure search.
The equivalent for graphs seems a little strange - how many subgraph searches would be useful? You would have to find a subgraph isomorphism to the graphs in the database.
For exact matches, canonicalising the input would be necessary for fast searching against the database. You could also store invariants for the graphs and use those to speed up the search. This is the concept of 'fingerprints' in computational chemistry.
