Cliques in an alternate graph representation
You haven't specified exactly what your output is, but if it is the standard "output all maximal cliques", then the size of your output is exponential in the worst case, and therefore regardless of what algorithm you use, or what your representation* is, the algorithm may have to spend exponential time, just to output the answer.

*This is assuming the number of cliques you can represent in a single machine word is at most polynomial (which it clearly is, for any reasonable model).