Abdi–Ghorbani's uniqueness conjecture for minimum-gap quartic graphs

Let $G$ be a connected quartic graph, meaning a $4$-regular graph, and let its spectral gap be the difference between its two largest adjacency eigenvalues. For each $n\geq 11$, let ${\cal F}_{\rm qua}$ be the specified subfamily of ${\cal F}_{\rm qua}^*$: write $n-11=5q+r$ with $q$ a non-negative integer and $0\leq r\leq 4$, take $q$ middle blocks $M$, and choose the two end blocks according to $r$ as described in the construction.

Abdi–Ghorbani's uniqueness conjecture. For every $n\geq11$, the $n$-vertex graph of ${\cal F}_{\rm qua}$ is the unique graph with minimum spectral gap among connected quartic graphs of order $n$.

The preceding theorem shows only that every minimum-gap connected quartic graph belongs to ${\cal F}_{\rm qua}^*$; this conjecture specifies the unique member of the smaller family ${\cal F}_{\rm qua}$ that should attain the minimum.