Fig. 9.

The discrete eigenvalue distributions $f_{(G)}$ of the amplifier constructors ($N_{\mathit{eff}}/N>1$) and $f_{(G^{\prime })}$ for the remaining graphs with ($N_{\mathit{eff}}/N<1$) and the smoothed spectral densities ${\phi }_{(\mathcal{G})}$ and ${\phi }_{({\mathcal{G}}^{\prime })}$ according to Eq. (12) for all quartic regular graphs with $N=14$, a–c, and all cubic regular graphs with $N=16$, d–f. The red lines in (b) and (e) depict the shape of the Kesten–McKay distribution (16), which is rescaled in order to cover the same area as the bars of the histogram. The spectral density of the amplifier constructors is shown as red line in (c) and (f), while the black line is for the remaining graphs. See also Fig. 15 of the Appendix for results of the smoothed spectral densities ${\phi }_{(\mathcal{G})}$ and ${\phi }_{({\mathcal{G}}^{\prime })}$ for the other N