Assess the student's ability to construct explicit infinite families of k‑regular Ramanujan graphs using quaternion algebras and automorphic representations. Tasks include: formulate the arithmetic/quaternionic setup (orders, reduction mod p, Cayley/quotient graphs); produce an explicit infinite family (e.g., LPS or function‑field analogues) and give generators; prove the Ramanujan eigenvalue bounds by relating adjacency spectra to Hecke eigenvalues via the Jacquet–Langlands correspondence and Deligne/Drinfeld bounds; compute spectra for sample primes and discuss parameter choices, limitations, and variations. Emphasis is on construction, rigorous spectral proof, and explicit examples suitable for graduate research.