The 1935 NCAA football season was the last one before the Associated Press writers' poll was used in selecting the national champion. The Dickinson System, consisting of the calculations of University of Illinois Professor Frank Dickinson, crowned Southern Methodist University (SMU) as the best in the nation. A poll of newspaper writers, taken at year's end—by United Press rather than the AP—concluded that the Minnesota was the best in the nation.
The 1935 season also marked the first time the Heisman Trophy was awarded. It was won by Jay Berwanger of Chicago. Taken from https://en.wikipedia.org/wiki/1935_college_football_season

Final Stats:

Conference	Best Team	Worst Team
Southwest Conference	Southern Methodist	Texas A&M
Western Conference	Minnesota	Wisconsin
Southeastern Conference	Louisiana State	Sewanee
Pacific Coast Conference	California	Idaho
Big Six Conference	Nebraska	Iowa State
Independent	Princeton	Cornell
Southern Conference	North Carolina	Virginia Military Institute
Border Intercollegiate Athletic Association	New Mexico State	Texas-El Paso
Missouri Valley Conference	Washington (MO)	Oklahoma State
Rocky Mountain Conference	Utah State	Western State