# ALGEBRAIC COMBINATORICS

Graph toughness from Laplacian eigenvalues
Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 53-61.

The toughness $t\left(G\right)$ of a graph $G=\left(V,E\right)$ is defined as $t\left(G\right)=min\left\{\frac{|S|}{c\left(G-S\right)}\right\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c\left(G-S\right)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t\left(G\right)$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.197
Classification: 05C42,  05C50,  05C70,  05C45
Keywords: Toughness, Laplacian eigenvalue, perfect matching, factor, Hamilton cycle
Gu, Xiaofeng 1; Haemers, Willem H. 2

1 Department of Computing and Mathematics University of West Georgia Carrollton, GA 30118, USA
2 Department of Econometrics and Operations Research Tilburg University Tilburg, The Netherlands
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Gu, Xiaofeng; Haemers, Willem H. Graph toughness from Laplacian eigenvalues. Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 53-61. doi : 10.5802/alco.197. https://alco.centre-mersenne.org/articles/10.5802/alco.197/

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