The signless Laplacian spectral Turán problems for disjoint triangles

The classical Turán-type problem for graphs forbidding $k$ disjoint triangles has been resolved firstly by Erdős [Arch. Math. 13 (1962) 222-227] for graphs of sufficiently large order and improved by Moon [Canad. J. Math. 20 (1968) 95-102], who established that the unique edge extremal graph is $K_{k-1}\vee T_2(n-k+1)$ for $n>\frac{19k-11}{2}$. In this paper, we determine the maximum signless Laplacian spectral radius among all $k{K}_3$-free graphs of order $n$ and characterize the unique signless Laplacian spectral extremal graph $K_{k-1}\vee T_2(n-k+1)$ for all $k\ge 2$ and sufficiently large $n$.