I'm interested in graph families which are sparse, and by sparse I mean the number of edges is linear in the number of vertices. |E| = O(|V|).
Besides non-trivial minor-closed families of graphs (these turn out to be sparse), I don't know any other families. Can anyone suggest any interesting graph families (which are not minor-closed) which are sparse?


Please don't suggest the trivial family ("the family of sparse graphs").


EDIT: Thanks to the first few people who replied, I realized that bounded degree graphs (max degree < k) also form an interesting and large class of sparse graphs. So perhaps I'll refine my question to exclude those too. Any interesting sparse graph families where the max degree isn't bounded? For example the family of star graphs is sparse and not bounded degree. (But they're minor-closed.)



Interesting families of sparse graphs?
1) Graphs of degree at most d. There are a myriad results known about them. Among them there are regular graphs (graphs in which every vertex is of degree exactly d).


2) Graphs of bounded degeneracy. These are the graphs in which every subgraph has a vertex of degree at most d. For example, stars have unbounded degree, but degeneracy 1. These share some common properties with graphs of bounded degree. For example, there are several results in Ramsey theory that have been extended from graphs of bounded degree to graphs of bounded degeneracy.


3) Random graphs of edge density c/n for constant c. Strictly speaking these are not a 'family of graphs', but since for properties of interest a random graph usually either satisfies it with high probability or fails it with high probability, it is fair to add them. There are also various other models of random graph that yield sparse graphs.


4) On a lower level, there are particular subfamilies of the above, such as stars, grids, etc. 


