It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc.
However, I am curious if there are any special properties of bipartite expanders that more general families of expanders don't have (or vice versa). In particular, are there any extreme differences, where bipartite and non-bipartite expanders differ greatly (especially in a combinatoric, algorithmic, or complexity-theoretic sense)?
A priori, we might expect that bipartite and non-bipartite expanders would likely share many pseudorandom properties, and in fact are constructible from each other. So that would suggest a negative answer to this question.
On the other hand, bipartite graphs in general have many special properties (eg. König's theorem) that have complexity-theoretic implications. So it's not unreasonable to think that differences between random bipartite and non-bipartite graphs may yield interesting differences between bipartite and non-bipartite expanders.
This is sort of a vague, open ended question, since I'm not exactly sure what kind of answer I'd like, and I am open to any interpretation. However, an example of a 'non-answer' might be 'bipartiteness is distinguished by the smallest eigenvalue in the spectrum'; I'm more interested in specifically bipartite expanders rather than spectral properties of bipartite graphs as a whole.