  In this paper, we use abstract Lyapunov graphs as a combinatorial tool to
obtain a complete classification of Smale flows on
$\mathbb{S}^2\times\mathbb{S}^1$. This classification gives necessary and
sufficient conditions that must be satisfied by an abstract Lyapunov graph in
order for it to be associated to a Smale flow on
$\mathbb{S}^2\times\mathbb{S}^1$.
