Inertia, Independence and Expanders

TL;DR

This paper explores the relationships between various graph parameters like independence number, Lovász theta, Shannon capacity, and eigenvalue counts, proving new bounds and conjectures using spectral expanders.

Contribution

It proves conjectures relating eigenvalue counts to independence and Shannon capacity, and introduces new bounds involving spectral expanders and the Lovász theta function.

Findings

Existence of graphs with small independence number and arbitrarily large eigenvalue counts.

Existence of graphs with Shannon capacity at most 3 and arbitrarily large eigenvalue counts.

Lovász theta can be exponentially larger than the eigenvalue count, refining previous bounds.

Abstract

Let $G$ be a graph on $n$ vertices, independence number $\alpha(G)$, Lov\'asz theta function $\vartheta(G)$, and Shannon capacity $\Theta(G)$. We define $n_{\ge0}(G)$ to be the minimum number of non-negative eigenvalues taken over all Hermitian weighted adjacency matrices of $G$. It is well known that $\alpha(G) \le \Theta(G) \le \vartheta(G)$ and $\alpha(G) \le n_{\ge0}(G)$. Continuing a long line of work, we investigate the relationships between $ \alpha(G) $, $ \vartheta(G) $, $\Theta(G)$, and $ n_{\ge 0}(G) $. We prove a conjecture of Kwan and Wigderson, showing that for every integer $k$, there exists a graph $G$ with $\alpha(G) \leq 2$ and $n_{\ge 0}(G) \ge k$. In addition, we prove that for every integer $k$, there exists a graph $G$ with $\Theta(G) \leq 3$ and $n_{\ge 0}(G) \ge k$. Both results rely on a new observation: if the complement of $G$ contains a good spectral expander, then $n_{\geq 0}(G)$ must be large. We also show that $\vartheta(G)$ can be exponentially larger than $n_{\ge 0}(G)$, improving a recent result of Ihringer.