Lyapunov graphs of nonsingular Smale flows on $S^1\times S^2$

In this paper, following J. Franks’ work on Lyapunov graphs of nonsingular Smale flows on $S^3$, we study Lyapunov graphs of nonsingular Smale flows on $S^1\times S^2$. More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1\times S^2$. We also study the singular type vertices in Lyapunov graphs of nonsingular Smale flows on 3-manifolds.