A recent dramatic increase in air travel has resulted in similarly dramatic increases in earnings for three major airlines - United, Trans World and Alleghany. Each reported yesterday record-breaking earnings for the second quarter.

United Airlines' parent company, UAL Inc., reported a 223 percent increase in second-quarter earnings, to $97.9 million ($3.93 a share) from $30.3 million ($1.21) a year ago.

Six-month profits climbed to a record $115.1 million ($4.61 a share) from $6.3 million (25 cents) in 1977. The nation's largest airline credited the advance primarily to the increased use of discount air fares, a result of actions by Civil Aeronautics Board.

Trans World Airlines Inc. reported a 71 percent increase in its second-quarter earnings, to $48.3 million ($2.78 a share) from $28.2 million ($1.92) a year ago. It was company's second best quarter in history, it said.

Earnings for the six months were $9.6 million (20 cents a share) compared with a loss of $22.8 million ($1.65) in 1977. TWA Chairman Edwin Smart said he expects 1978 to be "another year of record earnings."

Allegheny Airlines, based at National Airport, reported a 22.2 percent increase in second-quarter earnings, to $20.2 million ($3.24 a share) from $6.3 million (99 cents) a year ago.

Six months profits for the nation's sixth largest airline were also a record - $13.2 million ($2.13 a share) compared with $3.5 million (54 cents) in 1977.

Allegheny Chairman Edwin Colodny said the record earnings "can be attributed, in large part, to the continuing strength of consumer spending and popularity of our variety of low fare travelplans."