The club record signing was the top scorer in the Championship for Watford last season, but has now joined Swansea on a four-year deal for a £3.5million fee.
"I'm delighted to have got everything sorted,'' Graham told the club's official website. "I had been away on holiday so couldn't do it any sooner. Now I just can't wait to get started.
"I know Brendan has been an admirer of me for some time, which is flattering, and he had a bid rejected for me at the start of the season.
"But it's a new chapter for me now at Swansea and I'm looking forward to it and having a good season in the Premier League.
"I've had a good chat with the gaffer and I liked what he had to say. He's driven; I'm driven and I'm looking forward to working with him and establishing a good relationship with him, the team and the fans.
"It's going to be a difficult task because there are no easy games in the Premier League. But the squad at Swansea is very strong and the way they play is suited to the Premier League. Hopefully I can come in and add to that.
"The way Swansea play is how football should be played and was a massive factor, together with my chat with Brendan, to me coming here."
Regarding his price tag, Graham said: "There's pressure, there always is when it's a big amount like that. But there's not as much pressure on me as Andy Carroll and £35m!
"My job is to score goals and play well for the team - and that's what I'm looking to do. I want to score goals and I'm sure I will."