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).