The Algebraic Connectivity of a Graph and its Complement

TL;DR

This paper investigates the algebraic connectivity of a graph and its complement, providing bounds on their second smallest Laplacian eigenvalues and addressing a conjecture in spectral graph theory.

Contribution

It proves a new lower bound for the maximum algebraic connectivity of a graph and its complement, advancing understanding of spectral properties in graph complements.

Findings

Established that rac{2}{5} for the maximum of rac{2(G), 2(\u007f G)

Addressed a conjecture relating 2(G) and 2( G)

Contributed to spectral graph theory by bounding eigenvalues of graph complements.

Abstract

For a graph $G$, let $\lambda_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $\lambda_2(G) + \lambda_2(\overline G) \ge 1$, where $\overline G$ is the complement of $G$. In this paper, it is shown that $\max\{\lambda_2(G), \lambda_2(\overline G)\} \ge 2/5$.