Maybe ..by train.

I'd recommend the bus, it takes a little longer but it's generally cheaper and runs regularly. The bus station is a short walk from Prince's Street (the centre of town). I think the 'Citylink' website often does deals on bus fares. It's been a while since I've been in Edinburgh though.

The train fare between Glasgow and Edinburgh is ridiculously expensive!

I've taken the Citylink bus before from Edinburgh airport to Buchanan bus station in the centre of Glasgow. It's clean, comfortable and cheaper than the train. As Drewmentions, there's also a Citylink bus that leaves from the centre of Edinburgh.