Usually, one defines an expander graph to be a regular graph satisfying one of the following properties: Either the edge-expansion is large, or the spectral gap is large, or the mixing time is at most logarithmic in the number of vertices, or it satisfies a mixing-lemma type property.
It doesn't really matter much which property we require, since up to a constant they are all equivalent. This is also the reason why the concept of expander graphs is so powerful.
My question is: What happens in graphs that are far from regular? One can look at each of these properties separately, but is there a sense in which they are equivalent? If not, does it make sense to talk about expander graphs in this setting?