The signless Laplacian spectral radius of graphs without disjoint cliques

A graph $G$ is $(t+1)K_{r+1}$-free if it contains no $t+1$ pairwise vertex-disjoint copies of $K_{r+1}$.

Moon [Canad.

J.

Math.

20 (1968) 95-102] and Simonovits [Theory of Graphs (Proc.

Colloq., Tihany, 1966)] independently determined that, for sufficiently large $n$, $K_{t}\vee T_{r}(n-t)$ is the unique $n$-vertex $(t+1)K_{r+1}$-free graph with the maximum number of edges.

In 2023, Ni, Wang and Kang [Electron.

J.

Combin.

30 (2023) \#P1.20] showed that the graph $K_{t}\vee T_{r}(n-t)$ is also the unique adjacency spectral extremal graph over all $n$-vertex $(t+1)K_{r+1}$-free graphs for sufficiently large $n$.

In this paper, for $r\geq 3$ and $t\geq 0$, we prove that $K_{t}\vee T_r(n-t)$ is the unique graph attaining the maximum signless Laplacian spectral radius among all $(t+1)K_{r+1}$-free graphs of sufficiently large order $n$.