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 that every such graph $G$ satisfies $$\rho (G)\le \rho (H_{n,k}),H_{n,k}=K_{k-2}\vee (K_{n-k+1}\cup K_1),$$ with equality if and only if $G\cong H_{n,k}$. Equivalently, the unique extremal graph is obtained from $K_{n-1}$ by adding one vertex adjacent to exactly $k-2$ vertices of the clique. Our proof combines a decomposition lemma for sparse complements, derived from the Hajnal–Szemerédi theorem, with the Motzkin–Straus inequality and spectral estimates based on quotient matrices and the Rayleigh quotient.