Toughness and normalized Laplacian eigenvalues of graphs

TL;DR

This paper establishes new lower bounds for graph toughness based on degrees and normalized Laplacian eigenvalues, generalizing Brouwer's conjecture and characterizing extremal graphs.

Contribution

It introduces bounds linking toughness with spectral properties and characterizes graphs that attain these bounds, extending prior conjectures and results.

Findings

Lower bounds for toughness in terms of spectral data

Characterization of graphs achieving the bounds

Extension of Brouwer's toughness conjecture

Abstract

Given a connected graph $G$, the toughness $\tau_G$ is defined as the minimum value of the ratio $|S|/\omega_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $\omega_{G-S}$ is the number of connected components in the subgraph $G-S$ obtained by deleting all vertices of $S$ from $G$. In this paper, we provide a lower bound for the toughness $\tau_G$ in terms of the maximum degree, minimum degree and normalized Laplacian eigenvalues of $G$. This can be viewed as a slight generalization of Brouwer's toughness conjecture, which was confirmed by Gu (2021). Furthermore, we give a characterization of those graphs attaining the two lower bounds regarding toughness and Laplacian eigenvalues provided by Gu and Haemers (2022).