Maximal-gap conjecture for the spectra of the cubic graphs $\Delta_n$

Let $\Delta_n$ be the cubic graph in the family of minimum-spectral-gap cubic graphs for the relevant orders, and let the spectrum of $\Delta_n$ be the list of adjacency eigenvalues. A gap interval is an interval containing no eigenvalues of the sequence.

Maximal-gap conjecture for $\Delta_n$. The interval

$$(1,\sqrt{5}]$$

is a maximal gap interval for the sequence $\Delta_n$.

The paper proves that this interval contains no eigenvalues of $\Delta_n$; the conjecture asserts that it cannot be enlarged while remaining a gap interval.