Design an explicit parametrized infinite family {G_k : k ≥ 1} of connected graphs such that every pair of distinct graphs in the family is cospectral with respect to the adjacency matrix but non-isomorphic. Your submission should (a) give a clear construction and prove the family is infinite and each G_k is connected, (b) prove pairwise cospectrality (e.g. via characteristic polynomials, Godsil–McKay switching, equitable partitions, or closed-walk counts), and (c) prove pairwise non-isomorphism using graph invariants (degree sequences, subgraph counts, automorphism groups, or other arguments). Include at least the first three explicit examples and explain why the method produces arbitrarily many non-isomorphic cospectral graphs. State any standard theorems you use and justify all spectral/non-spectral claims.