  We give lower bounds on the size and total size of clique partitions of a
graph in terms of its spectral radius and minimum degree, and derive a spectral
upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal
graphs attaining the bounds are exactly the block graphs of Steiner $2$-designs
and the regular graphs with $K_t$-decompositions, respectively.

Combinatorics