On Eigenvalue Bounds for Bounded Genus Graphs and Minor-Free Graphs

Abstract

In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $\lambda_2(L_G)\lesssim\Delta\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $\lambda_2(L_G)\lesssim\Delta\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lov\'asz local lemma.