In this survey we present the interplay between topological dynamical systems theory with network flow theory in order to obtain a continuation result for abstract Lyapunov graphs $L(h_0, \ldots, h_n,\kappa)$ in dimension $n$ with cycle number $\kappa$. We also show that an abstract Lyapunov graph satisfies the Poincar\'{e}-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number $\gamma_1$ is greater than or equal to $\kappa$. We define the Morse polytope determined by the Morse inequalities and describe some of its geometrical properties.