On the Fiedler value of large planar graphs

TL;DR

This paper investigates the maximum algebraic connectivity (Fiedler value) of large planar graphs and related classes, establishing bounds that improve understanding of spectral properties in graph theory.

Contribution

It provides new bounds on the maximum Fiedler value for planar graphs and extends these bounds to bipartite, outerplanar, bounded genus, and $K_h$-minor-free graphs.

Findings

Bounds on $oxed{ ext{maximum Fiedler value}}$ for planar graphs.

Bounds for bipartite and outerplanar graphs.

Almost tight bounds for bounded genus and $K_h$-minor-free graphs.

Abstract

The Fiedler value $\lambda_2$, 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 $\lambda_{2\max}$, and we show the bounds $2+\Theta(\frac{1}{n^2}) \leq \lambda_{2\max} \leq 2+O(\frac{1}{n})$. 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 $\lambda_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.