TX Birth Idx:
Graham, Edward III, b 21 Jan 1932, Tarrant Co, TX. Mother Nora Sublett.
Her name is less common and I could find her on the 1930 census still at home, so she didn't get married until after that.
1930 census Tarrant Co, TX:
Sublett, Charles A., h, 73, wid, b TX, TN, MO
" John, s, 38, sgle, b TX, TX, TN
" Ella, d, 28, sgle, b TX
" Lilly, d, 26, sgle, b TX
" Frank, s, 24, sgle, b TX
" Elizabeth, d, 21, b TX
" Nora, d, 19, b TX ************
" Carrie, d, 16, b TX
I'm guessing they were married in Tarrant Co, TX. Maybe one of the researchers who browse this forum and has access to more sources than I do, can find that marriage for you.
There is a Cory Graham, 69, wid, born TX living with her married daughter and son-in-law just a few pages from the Sublett family on the 1930 Tarrant Co, TX census. Maybe she's the mother of your Edward????
Happy Hunting (^_^)