The largest airline brand can be categorized in many ways. They can be categorized based on the size of their aircraft, number of passengers carried, revenue, profit made, kilometers traveled and the number of employees.
According to wikipedia, based on profit, number of fleets, passengers carried and largest passenger kilometers flown, American Airline is the largest. In 2017 the airline carried about 200 million passengers. They have 950 fleet.
Lufthansa is the largest based on the number of employees. In 2017, Lufthansa had the highest revenue of about $42.16 billion.
By assets value and capitalization, Delta Airline is the largest.
FedEx is the largest by scheduled freight tonnes travelled in kilometers.