Extremal spectral gap of regular graphs with bounded vertex connectivity

The well-known inequality of Fiedler ensures that the spectral gap $r-λ_2(G)$ of a connected non-complete $r$-regular graph $G$ is bounded from above by the vertex connectivity $κ(G)$ of $G$. We prove that, for integers $t\geq2$ and $r>2t^2$, every connected $r$-regular graph with vertex connectivit