Toughness and distance spectral radius in graphs

Let G be a connected graph with vertex set V(G). A variation τ(G) of toughness of G, proposed by Enomoto in 1988, is defined as τ(G) = minS ⊂ V(G) { |S| / (c(G − S) − 1) : c(G − S) > 1 },where c(G − S) is the number of components of G − S for a subset S of V(G). For a positive real number τ, G is called τ-tough if τ(G) ≥ τ. Chen et al. (2024) gave sufficient spectral radius conditions for a graph G to be τ-tough, where the spectral radius is the maximum eigenvalue of the adjacent matrix of G and τ or 1/τ is a positive integer. Analogously, in this paper, we investigate the case of the distance spectral radius of G and present three tight distance spectral radius conditions, respectively, and analyze the extreme cases, which are similar to Chen et al.’s results.