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Currently, flight delays are common and they propagate from an originating flight to connecting flights, leading to large disruptions in the overall schedule. These disruptions cause massive economic losses, affect airlines' reputations, waste passengers' time and money, and directly impact the environment. This study adopts a network science approach for solving the delay propagation problem by modeling and analyzing the flight schedules and historical operational data of an airline. We aim to determine the most disruptive airports, flights, flight-connections, and connection types in an airline network. Disruptive elements are influential or critical entities in an airline network. They are the elements that can either cause (airline schedules) or have caused (historical data) the largest disturbances in the network. An airline can improve its operations by avoiding delays caused by the most disruptive elements. The proposed network science approach for disruptive element analysis was validated using a case study of an operating airline. The analysis indicates that potential disruptive elements in a schedule of an airline are also actual disruptive elements in the historical data and they should be considered to improve operations. The airline network exhibits small-world effects and delays can propagate to any part of the network with a minimum of four delayed flights. Finally, we observed that passenger connections between flights are the most disruptive connection type. Therefore, the proposed methodology provides a tool for airlines to build robust flight schedules that reduce delays and propagation.