Spectral extremal graphs for disjoint cliques

Abstract: The $kK_{r+1}$ is the union of $k$ disjoint copies of $(r+1)$-clique. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then $K_{k-1}\vee T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper, we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We prove that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to $K_{k-1}\vee T_{n-k+1,r}$.