  The inverse eigenvalue problem of a graph $G$ aims to find all possible
spectra for matrices whose $(i,j)$-entry, for $i\neq j$, is nonzero precisely
when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is
completely solved for a subfamily of clique-path graphs, in particular for
lollipop graphs and generalized barbell graphs. For a matrix $A$ with
associated graph $G$, a new technique utilizing the strong spectral property is
introduced, allowing us to construct a matrix $A'$ whose graph is obtained from
$G$ by appending a clique while arbitrary list of eigenvalues is added to the
spectrum. Consequently, many spectra are shown realizable for block graphs.
