All graphs with at most three nodes are C_4-free, so a(n) = 1 for n <= 3 and any graph is optimal.
For 4 <= n <= 9, the following are all optimal graphs, i.e., graphs that have n nodes and a(n) maximal C_4-free subgraphs:
n = 4: the 4-cycle;
n = 5: K_{2,3};
n = 6: the prism graph and the octahedral graph;
n = 7: the complement of 2*K_2 + K_3;
n = 8: K_4 X K_2 (Cartesian product) and the 16-cell;
n = 9: the circulant graph C_9(1,3), and K_{3,3,3} with three edges removed, one edge between the first and second parts in the partition and two edges from two other nodes in these two parts to a node in the third part.
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