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arXiv · 1206.3870

On the Fiedler value of large planar graphs

Lali Barrière·Clemens Huemer·Dieter Mitsche·David Orden
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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.

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Lali Barrière, Clemens Huemer, Dieter Mitsche, David Orden. 2012-06-18. On the Fiedler value of large planar graphs. https://doi.org/10.1016/j.laa.2013.05.032

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