Test the ability to design explicit infinite families of d-regular expander graphs and produce rigorous spectral-gap (Cheeger) bounds: students should give concrete constructions (e.g., Cayley/Ramanujan graphs, zig-zag or graph-product methods), specify parameters proving d-regularity, compute or bound the second-largest eigenvalue, relate eigenvalue bounds to combinatorial expansion via Cheeger inequalities, and present complete proofs of the claimed spectral-gap constants and asymptotic behavior.