  The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple
graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are
joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in
\mathbb{N}$. Here we determine the characteristic polynomial of
$\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any positive integers $m$
and $n$. Additionally, for some particular values of $m$ and $n$, we simplify
the above characteristic polynomials and provide the full spectrum in a few
cases.
