Assess the ability to construct explicit infinite families of d-regular graphs (fixed d≥3) with prescribed spectral-gap properties, to derive and rigorously prove tight upper and lower bounds on nontrivial eigenvalues (e.g., second-largest in absolute value), and to relate those bounds to Alon–Boppana and Ramanujan benchmarks. Require explicit constructions (for example Cayley/LPS/Morgenstern graphs, zig‑zag products, or lifts), adjacency/operator computations, asymptotic eigenvalue estimates, proofs of tightness, and at least two concrete family examples with full spectral analysis.