The improved diameter and order bounds for cubic graphs

Let $G$ be a cubic graph, and let $D$ be its diameter. The cubic-graph algebraic-connectivity bound conjecture. Its algebraic connectivity satisfies

$$\lambda_{2}(G)\leq 3-2^{3/2}\cos\left(\frac{\pi}{D}\right).$$

Moreover, if $G$ has order $n=2^{K+1}-2$, then

$$\lambda_{2}(G)\leq 3-2^{3/2}\cos\left(\frac{\pi}{K}\right).$$

These conjectured bounds are numerical improvements over the cited Nilli bound and are posed as open conjectures in the source.