Disruptions in airline operations are common, especially during extreme weather events, such as hurricanes. These phenomena can and do interrupt smooth and efficient passenger transportation, and severely affect and halt airline operations. Airport network topology plays a crucial role in the development of tools to implement efficient and effective recovery actions for different stakeholders during a disruption. Hence, in this research, we propose a graph theoretical approach that studies the flight/airport network from a topology perspective, but that also incorporates a geospatial component. In our approach, we present two mathematical frameworks to help classify airports in an airport network from two perspectives: their extent of getting disrupted and being a subject to the effect of disruption. We employ simple paths of certain lengths in three different ways to reveal the weather-related impact based on airport position as a direct destination, an intermediate connection, or due to its geographical location in relation to the extreme weather event. Additionally, we develop a rerouting method to identify potential airports for redirecting passengers from a hurricane impacted airport. The computational results show the significance of incorporating geospatial element in identifying the most affected airports during a weather disruption. Moreover, our experimental analysis shows that the forecast track of a hurricane and its impact distance affect the airport network disruption. All of our codes and data are publicly available at https://github.com/harshithameda/AnalysisNetworkDisruptions.