Connected graphs with minimum adjacency spectral gap

Let $G$ be a connected graph, and let $λ_1(G) > λ_2(G)$ denote its two largest adjacency eigenvalues. The spectral gap of $G$ is defined as the difference $λ_1(G) - λ_2(G)$. For integers $r\geq 2$ and $s\geq 0$, the double kite $DK(r,s)$ is formed by taking two vertex-disjoint copies of the complete