  Let $G$ be connected uniform hypergraph and let $\mathcal{A}(G)$ be the
adjacency tensor of $G$. The stabilizing index of $G$ is the number of
eigenvectors of $\mathcal{A}(G)$ associated with the spectral radius, and the
cyclic index of $G$ is the number of eigenvalues of $\mathcal{A}(G)$ with
modulus equal to the spectral radius. Let $G_1 \odot G_2$ and $G_1 \Box G_2$ be
the coalescence and Cartesian product of connected $m$-uniform hypergraphs
$G_1$ and $G_2$ respectively. In this paper, we give explicit formulas for the
the stabilizing indices and cyclic indices of $G_1 \odot G_2$ and $G_1 \Box
G_2$ in terms of those of $G_1$ and $G_2$ or the invariant divisors of their
incidence matrices over $\mathbb{Z}_m$, respectively.
