Distance signless Laplacian spectral radius and tough graphs involving minimun degree

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of G, respectively. The number of components of G is denoted by $c(G)$. Let t be a positive real number, and a connected graph G is t-tough if t c(G-S)≤|S| for every vertex cut S of $V(G)$. The toughness of graph G, denoted by τ(G), is the largest value of t for which G is t-tough. Recently, Fan, Lin and Lu [European J. Combin. 110(2023), 103701] presented sufficient conditions based on the spectral radius for graphs to be 1-tough with minimum degree δ(G) and graphs to be t-tough with t≥ 1 being an integer, respectively. In this paper, we establish sufficient conditions in terms of the distance signless Laplacian spectral radius for graphs to be 1-tough with minimum degree δ(G) and graphs to be t-tough, where 1/t is a positive integer. Moreover, we consider the relationship between the distance signless Laplacian spectral radius and t-tough graphs in terms of the order n.