A proof of Haemers' toughness conjecture

TL;DR

This paper proves a lower bound on the toughness of connected graphs based on Laplacian eigenvalues, confirming Haemers' conjecture.

Contribution

It establishes a spectral bound on graph toughness, providing a proof for Haemers' toughness conjecture.

Findings

Toughness is bounded below by the ratio of the second Laplacian eigenvalue to the difference between the largest eigenvalue and minimum degree.

The result confirms Haemers' conjecture relating spectral properties to graph toughness.

The proof applies to connected graphs with specified Laplacian eigenvalues.

Abstract

We prove that if $\Gamma$ is a connected graph with minimum degree $\delta$ and Laplacian eigenvalues $0=\mu_1<\mu_2\leqslant \cdots \leqslant \mu_n$, then the toughness of $\Gamma$ is bounded below by $\mu_2/(\mu_n-\delta)$.