Graph toughness from Laplacian eigenvalues

TL;DR

This paper establishes new lower bounds for graph toughness using Laplacian eigenvalues, supports a conjecture for improved bounds, and derives applications related to perfect matchings, factors, and Hamiltonicity.

Contribution

It introduces two tight lower bounds for graph toughness based on Laplacian eigenvalues and supports a conjecture that could unify and improve existing bounds.

Findings

Derived two tight lower bounds for toughness from Laplacian eigenvalues.

Supported a conjecture that, if true, would unify and improve existing bounds.

Obtained new results on perfect matchings, factors, and Hamiltonicity from Laplacian eigenvalues.

Abstract

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalues are obtained, which leads to a conjecture about Hamiltonicity and Laplacian eigenvalues.