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Homeomorphically irreducible trees
These are trees with no vertices of degree 2. They are also called seriesreduced trees. The following collection was calculated by Brendan McKay (source) and lists all homeomorphically irreducible trees on up to 28 vertices.

Tetravalent graphs
Tetravalent, quartic or 4regular graphs are graphs where every vertex has degree 4. The following collection was calculated by The House of Graphs (source) and lists all connected tetravalent graphs on up to 15 vertices.

Uniquely hamiltonian graphs
A uniquely hamiltonian graph is a graph which contains exactly one hamiltonian cycle. The following collection was calculated by The House of Graphs (source) and lists all uniquely hamiltonian graphs on up to 10 vertices.

Minimal Cayley graphs
A Cayley graph is a graph that encodes the abstract structure of a group. An undirected graph G is a minimal Cayley graph if there is a group Γ with an inclusionminimal inverseclosed generating set C such that the Cayley graph Cay(Γ,C) is isomorphic to G. The following collection was calculated...

Perihamiltonian graphs
A graph is perihamiltonian if it is nonhamiltonian but all its edgecontracted subgraphs are hamiltonian. The following collection was calculated by the House of Graphs (source) and lists all perihamiltonian graphs on up to 12 vertices.