# Graph Toughness from Laplacian Eigenvalues

Xiaofeng Gu, Willem H. Haemers

Research output: Working paperOther research output

## 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.
Original language English Ithaca Cornell University Library 11 Published - 8 Apr 2021

### Publication series

Name arXiv 2104.03845

## Keywords

• math.CO
• 05C42, 05C50, 05C70, 05C45

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