Toughness and normalized Laplacian eigenvalues of graphs

Given a connected graph $G$, the toughness $τ_G$ is defined as the minimum value of the ratio $|S|/ω_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $ω_{G-S}$ 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 $τ_G$ 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).

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