Gap sets for the spectra of regular graphs with minimum spectral gap

Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more `symmetric' compared to the other graphs in these two families, respectively. We prove that $(1,\sqrt{5}]$ is a gap interval for $\Delta_n$, and $[(-1+\sqrt{17})/2,3]$ is a gap interval for $\Gamma_n$. We conjecture that these two are indeed maximal gap intervals. As a by-product, we show that the eigenvalues of $\Delta_n$ lying in the interval $[-3,-\sqrt{5}]$ (in particular, its minimum eigenvalue) converge to $(1-\sqrt{33})/2$ and the eigenvalues of $\Gamma_n$ lying in the interval $[-4,-(1+\sqrt{17})/2]$ (and in particular, its minimum eigenvalue) converge to $1-\sqrt{13}$ as $n$ tends to infinity. The proofs of the above results heavily depend on the following property which can be of independent interest: with few exceptions, all the eigenvalues of connected cubic and quartic graphs with minimum spectral gap are simple.