A Nordhaus--Gaddum problem for the spectral gap of a graph

TL;DR

This paper investigates bounds on the spectral gap of a graph and its complement, establishing conditions under which the spectral gap is bounded below by a constant or diminishes with the number of vertices.

Contribution

It proves new bounds on the spectral gap for a graph and its complement, including conditions for constant and diminishing spectral gaps, extending Nordhaus-Gaddum type results.

Findings

Spectral gap of G and its complement is at least Omega(1/n).

If degrees are proportional to n, the spectral gap is Theta(1).

Constructs graphs with spectral gap O(1/n^{3/4}) for both G and its complement.

Abstract

Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of $G$ and $\overline{G}$ is $\Omega(1/n)$. Moreover, if all degrees are $\Omega(n)$ and $n-\Omega(n)$, then the larger spectral gap of $G$ and $\overline{G}$ is $\Theta(1)$. We also show that if the maximum degree is $n-O(1)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $\Omega(1/n)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\overline{G}$ is $O(1/n^{3/4})$.