The kK_{r+1}-free n-vertex graph with maximum t-clique spectral radius is K_{k-1} joined to T_r(n-k+1) for sufficiently large n.
Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl., 432 (2010), 2243–2256
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Generalized spectral Tur\'an problems for disjoint cliques
The kK_{r+1}-free n-vertex graph with maximum t-clique spectral radius is K_{k-1} joined to T_r(n-k+1) for sufficiently large n.