A quick numerical investigation for $N$ up to 2048 leads to a conjectured expansion of the largest eigenvalue, $$ \lambda_0(N) = 4 - \frac{2\pi}{N} + \frac{\pi^2}{2 N^2} - \frac{\pi^3}{24 N^3} + \frac{\pi^4}{48 N^4} - c_5 \frac{\pi^5}{N^5} + \ldots. $$ The coefficient $c_5 \approx 1.3155$ seems to be a more complicated fraction. I don't think that the problem simplifies for $N$ being a power of two.
Fred Hucht
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