I've been writing a little about some results on graph theory, and I want some nice examples of applying the results to some interesting finite connected graphs to show how the results might be useful. I'm therefore looking for a reference which has a collection of interesting graphs, ideally along with a picture if possible (but not the end of the world if there are no pictures) and some of the basic properties of the graph; for example, connectedness, diameter etc, though again if the reference only has the names I'm sure I can track down their properties online so that isn't too big a problem either.
I have been using Wikipedia's gallery of named graphs which has been quite helpful, but unfortunately a lot of those turn out to be trivial cases for the results I'm proving, and Alain Matthes had an excellent list too (altermundus.fr/downloads/documents/NamedGraphs.pdf). Some of the examples I am interested in are nontrivial graphs with moderately large girth (say 8 or above, and nontrivial meaning not a path, tree, cycle etc), Cayley graphs, and any interesting constructions of sequences of graphs of increasing order $n \to \infty$. Other than such sequences I would prefer to look at examples of 'small to medium' order (so probably 60 or less), and on the less dense end of the scale; essentially graphs which you can feasibly work with "by hand" because they are not overly enormous, not graphs like the Higman-Sims.
I have tried to explain the sort of graphs I am looking for just in case it's relevant, but if there is an all-encompassing handbook or compendium of some sort then I am obviously very happy to sift through them myself to locate some useful examples. Any responses would be appreciated, be they books, websites, papers, or just individual suggestions of interesting graphs which weren't in the Wikipedia gallery. Thankyou!