I have been doing some searches myself to find the connection. Here is what I have found so far.
Frank Edwin "TUG" McGraw JR. is the father of Tim McGraw
TUG MCGRAW's Father
Frank E. McGraw SR.
Mother's maiden name: Mckenna
Social Security Death Index
about Frank E. McGraw JR.
Name: Frank E. McGraw JR.
Last Residence: 19312 Berwyn, Chester, Pennsylvania, United States of America
Born: 30 Aug 1944 Contra Costa, California
Died: 5 Jan 2004
State (Year) SSN issued: California (1960 )http://en.wikipedia.org/wiki/Tug_McGraw
Name: MCGRAW, FRANK EDWIN SR.
Social Security #: 554031563
Birth Date: 21 Jul 1911
Death Date: 24 Jul 1991
Death Place: NAPA
Mother's Maiden Name: NAVE
1920 census Erie Ward 5, Erie, Pennsylvania
Edward J McGraw 47 abt 1873 Born in PA. Mom & Dad both born in Pennsylvania
Emma C McGraw 42 abt 1878 Born in PA. Mom born in VA. Dad born in MD. POSSIBLE MAIDEN NAME was NAVE.
Dorothy E McGraw 10 abt 1910 Born in PA.
Frank E McGraw 6 abt 1914 Born in MD.
William H McGraw 10 months abt 1919 Born in PA.
1910 census Erie Ward 2, Erie, Pennsylvania
Edward J. McGraw 36
Emma C. McGraw 32
Dorothy E. McGraw 1 month
Hope this helps
MIKE MCGRAW (KING GEORGE, VA, USA)