Graham was 79 and had been ill for some time.
He was the great-grandson of Thomas Graham, who co-founded the Express & Star newspaper in the 1880s.
His funeral was held last week, and the Express & Star published a special editorial section to commemorate his life and achievements.
“He was a gentleman publisher of the old school, a traditional newspaperman with ink in his veins, but also a moderniser who took both pride and joy in his newspapers,” the tribute said.
“He loved the crafts of printing and journalism, the excitement of the deadline and the rumbling of giant presses.”
Graham was praised for his pioneering use of colour printing when, aged just 20, he was tasked with launching the Shropshire Star, which was printed in colour more than 20 years before national newspapers adapted to colour production.
In the 1980s the Express & Star also became the first newspaper to introduce a computerised editorial system.
Graham is survived by his wife Angela and four children, with his sons Tom and Edward also working at the group.
The family-owned Claverley Group includes MNA Media, MNA Digital, the Guiton Group, PCS, Precision Colour Printing, Kennedy Publishing and Cubiquity.