Take a periodic one-dimensional lattice of size $N$ with $2k$ nearest neighborers. That is, vertex $i$ is connected to $i+1,i+2,...,i+k$ and $i-1,i-2,...i-k$ (with the understanding that the indices are modulo $N$). Call the associated adjacency matrix $A_k$, and consider the limit of large $N$.
If I'm not mistaken, the eigenvalue spectrum for $A_1$ can be well approximated by a cosine wave with amplitude $2$ and period $2N$.
Why does the spectra for $A_2$ have a 'kink'?
Why does the kink in $A_2$ appear at a magnitude of zero?
Why do the kinks in $A_3$ appear touch exactly when the curve hits the spectra for $A_2$ (it looks that after a certain point, one graph is supported by the other).
In general, the spectra for $A_k$ seem to have $k-1$ kinks, can this be shown?