Abstract: For any given simple graph G, the question of how to judge whether it has some structural properties has been explored by graph theory scholars. Because the spectrum of a graph can well reflect the structural properties of a graph and is easy for calculation, in recent years, many scholars have used the spectrum theory to study the related properties of a graph. In this paper, we first find the stability of the corresponding structural properties of graphs, then construct the corresponding closures of graphs; finally, by using the method of reductio ad absurdum, according to the signless Laplacian spectral radius of its complement, graph G with larger minimum degrees is the sufficient condition for s-connected, s-edge-connected, s-path-coverable, s-Hamiltonian, s-edge-Hamiltonian, s-Hamilton-connected or α（G）≤s.

Key words: signless Laplacian spectral radius, stability, closure, minimum degree