We consider partially-connected $K$-user interference networks, where the transmitters have no knowledge about the channel gain values, but they are aware of network topology (or connectivity). We introduce several linear algebraic and graph theoretic concepts to derive new topology-based outer bounds and inner bounds on the symmetric degrees-of-freedom (DoF) of these networks. We evaluate our bounds for two classes of networks to demonstrate their tightness for most networks in these classes, quantify the gain of our inner bounds over benchmark interference management strategies, and illustrate the effect of network topology on these gains.