On the Fiedler value of large planar graphs

The Fiedler value $λ_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $λ_{2\max}$, and we show the bounds $2+Θ(\frac{1}{n^2}) \leq λ_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $λ_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.

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