The 66-year-old from Culmstock says arresting the plummeting attendances at Ladysmead is one of the biggest challenges he will face while in charge of the Yellows.
But Graham says he is already forming a plan to deal with such problems, as he bids to bring the glory days of the 1990s back to the club.
The new chairman said: "We need to go and re-establish the club as a family club, the way it was when it was such a successful club in the 1990s.
"It started off as a family club with local people. There were lots of volunteers, lots of help and lots of loyal supporters.
"We really have tested the loyalty of supporters with our recent results. Last season was our poorest for a very long time.
"We want to get the social side going, get the income on the way up and do everything possible to entice the supporters to come back and watch Tiverton."
Graham, who has run Dobles Garage in Hemyock for about 30 years, has been on the committee at Tiverton for the past four years and recently enjoyed a short stint as vice-chairman.
He takes over the chairmanship from Dave Wright, who has stepped down into the vice-chairman's role after about 14 years at the helm.
Graham revealed his predecessor had signalled his intention to step down when the Yellows finished in a relegation spot last season, but manager Martyn Rogers' subsequent resignation meant Wright agreed to stay on until a new boss was found.
Graham said: "After last season David felt it was time for a change. He felt similar to how Martyn Rogers felt.
"At the end of the season David was expressing that he would step down, but the actual announcement was delayed because we then received Martyn's resignation.
"He was here for 14 or 15 years, so it's a fairly big challenge for me to follow on from him."