The planar graph algebraic-connectivity conjecture

Let $G$ be a planar graph, and let $a(G)$ denote its algebraic connectivity, the second-smallest eigenvalue of its Laplacian. The graphs $K_4$ and $2K_1\dot{+}C_4$ are the complete graph on four vertices and the join of two isolated vertices with a four-cycle, respectively. Planar algebraic-connectivity conjecture.

$$a(G)\leq 4,$$

with equality if and only if $G=K_4$ or $G=2K_1\dot{+}C_4$. If $G$ is neither of these graphs, then $a(G)\leq 3$.

The conjecture would determine the maximum algebraic connectivity of planar graphs and its equality cases; the source reports that it has not been proved.