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Chordal graphs
A chordal graph is a simple graph in which every graph cycle of length four and greater has a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. The following collection was calculated by Brendan McKay (source) and lists all connected chordal graphs up...

Countries of the world
This collection contains networks of countries grouped by region. Additionally, each graph represents a certain statistic about the country, ranging from population size, birthrate to coastline length and GDP.
The graphs were generated from a Kaggle dataset created by Fernando Lasso and... 
1000 popular IMDB movies
This collection contains networks of 1000 popular movies from 2006 to 2016 on the internet movie database or (IMDB) grouped by genre. Each network represents the connections between each movie and its top four actors in a certain genre. This generates interesting connected subgraphs, since most...

Edge4critical graphs
An edge4critical graph, is a graph that is connected, is not (vertex) 3colourable, and Ge is 3colourable for every edge e. The following collection was calculated Brendan McKay (source) and lists all connected edge4critical graphs on up to 13 vertices.

Arctransitive graphs
An arctransitive graph is a graph whose graph automorphism group acts transitively on its graph arcs. The following collection was calculated by Marston Conder (source) and lists all connected arctransitive graphs of order 2 to 30.