Assess students' ability to design an explicit family of finite d-regular expander graphs, prove a nontrivial spectral gap lower bound independent of graph size, and rigorously analyze connectivity, diameter (e.g., O(log n) bounds), and combinatorial/edge expansion using spectral and combinatorial methods (e.g., eigenvalue estimates, Cheeger-type inequalities, Cayley graph or zig-zag constructions); require clear statements of assumptions, derivation or bounding of the second-largest eigenvalue, proofs of connectivity and expansion constants, and discussion of optimality and applications.