On the Fiedler value of large planar graphs

The Fiedler value λ2, 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 λ2max, and we show the bounds 2+Θ(1n2)⩽λ2max⩽2+O(1n). 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 λ2max for two more classes of graphs, those of bounded genus and Kh-minor-free graphs.