# Non-Cayley expander graphs

When I search about expander graphs in google I see a lot of articles about expander Cayley graphs. Now my questions are as follows:

Are all expander regular graphs are Cayley, or there is a special case of connected prime-order expander regular graphs which are not Cayley (specially graphs that constructed from disjoint cycles)?

• If we take an infinite family of finite quotients of $G=SL_3(\mathbf{Z})$, we get expander graphs (which are Cayley). If we fix a finite subgroup $F$ of $G$, and we mod out these by $F$, it sounds likely that we get non-Cayley examples (even with $F$ of order 2); possibly these are even not vertex-transitive graphs.
– YCor
Apr 25 '16 at 0:37

The answer to your question is yes. The existence of such expanders which proved by probabilistic method, give us an optimistic view for search to find non-Cayley expanders. But, a first class of such graphs are $k$-regular bipartite which has some parallel edges. So, for your question these are not interesting. There is a general method which you can use for construction non-Cayley expander from an other family of expanders. The only key in this way, is preserving the below bound for the Cheeger constant. For more results and updated things, you can see these two resources, specially the section 3.1 of the second one: