We present FlipDyn-G, a dynamic game model extending the FlipDyn framework to a graph-based setting, where each node represents a dynamical system. This model captures the interactions between a defender and an adversary who strategically take over nodes in a graph to minimize (resp. maximize) a finite horizon additive cost. At any time, the FlipDyn state is represented as the current node, and each player can transition the FlipDyn state to a node based on the connectivity from the current node. Such transitions are driven by the node dynamics, state, and node-dependent costs. This model results in a hybrid dynamical system where the discrete state (FlipDyn state) governs the continuous state evolution and the corresponding state cost. Our objective is to compute the Nash equilibrium of this finite horizon zero-sum game on a graph. Our contributions are two-fold. First, we model and characterize the FlipDyn-G game for general dynamical systems, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for scalar linear discrete-time dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the system. Additionally, for a finite state birth-death Markov chain (represented as a graph) under scalar linear dynamical systems, we derive analytical expressions for the NE takeover strategies and saddle-point values. We illustrate our findings through numerical studies involving epidemic models and linear dynamical systems with adversarial interactions.