Spectral Extremal Graphs without a $K_k$-Factor

Let $k\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\ge 2k-1$. More precisely, we prove