A note on Fiedler value of classes with sublinear separators

TL;DR

This paper investigates the maximum second eigenvalue (Fiedler value) of graphs within certain classes, relating it to shallow minors and determining its order for minor-closed classes like planar graphs.

Contribution

It establishes a connection between the Fiedler value and shallow minors, providing the order of the Fiedler value for minor-closed classes such as planar graphs.

Findings

Determines the order of the Fiedler value for planar graphs.

Relates Fiedler value to shallow minors.

Provides bounds for classes with sublinear separators.

Abstract

The $n$-th Fiedler value of a class of graphs $\mathcal C$ is the maximum second eigenvalue $\lambda_2(G)$ of a graph $G\in\mathcal C$ with $n$ vertices. In this note we relate this value to shallow minors and, as a corollary, we determine the right order of the $n$-th Fiedler value for some minor closed classes of graphs, including the class of planar graphs.