Margulis's expanders: This class of 8 regular graphs is the first explicit example for a family of expanders. The vertices are pairs of integer modulo m, the neighbors of (x,y) are (x+y,y), (x-y,y), (x,y+x), (x,y-x), (x+y+1,y), (x-y+1,y), (x, y+x+1), (x,y-x+1). All operations are modulo m.
Expanders were first doscovered and constucted probabilistically by Pinsker. The Ramanujan graphs of Lubotzky, Philips and Sarnak are expanders with extremely good properties. This paper by Hoory, Linial and Wigderson contains much more information.
