Graham has spent nine years in the NFL with the Broncos and the Patriots. Although he has never been a star he has been solid and steady.
In his career he has accumulated 222 receptions, 2,465 receptions, and 24 touchdowns. He has also won two Super Bowls.
A proven winner and steady winner is definitely what the Titans need. With two young, but talented tight ends Jared Cook and Craig Stevens.
Cook is a fast athletic player who plays more like a wide receiver, but has trouble blocking, while Stevens is a big blocking guy who still needs to work on his catching and getting open.
A veteran guy like Graham who has experience in all aspects of the game can help both of these players reach their potential and help the Titans on their climb out of the NFL cellar.
Graham can also contribute as a third tight end. Although his athletic ability has diminished Graham can still contribute as a blocker and receiver.
Since Graham is not a premier tight end and his skills have diminished he will come at a relatively cheap price. The Titans have a lot of cap room and signing Graham wouldn't take up much of that room.
Since Graham is cheap and will help the team their is no reason to not sign him. It may not be a sexy signing, but it will serve it's purpose.