The Hoops' Glasgow enemies were relegated to the Third Division in the summer after being found guilty of financial breaches with some critics suggesting that it has affected the green and white side of the city.
Celtic currently sit fifth on the SPL table with two wins from five outings following the weekend's loss to St Johnstone but Mulgrew does not believe that the absence of 'Gers as title contenders has had any bearing on their mediocre beginning to their championship defence.
"I wouldn't say that," was Mulgrew's reply when asked by reporters if this was the case.
"We just didn't turn up and it wasn't good enough."
Neil Lennon's Celtic outfit next meet Portuguese club Benfica in their first group stage fixture of the Champions League at Celtic Park on Wednesday night prior to their league clash with Dundee on Saturday.