After more time than it was worth looking for a good (and free) API provider for college schedules, I ended up making a scraper to harvest them from ESPN. After the needed info was in my database and the search functions prepared, I had my answer.
Apparently, assuming that a head-to-head match is considered as 1, the maximum number of degrees of separation between any two FBS schools (this season) is 4.. Details as they pertain to Ole Miss after the jump.The only school with 4 degrees of separation from Ole Miss this season is Eastern Michigan:
|vs||New Mexico State|
I linked each of the schools images and locations to their respective schedules over at ESPN. The images are from ESPN's CDN site.
Ole Miss was a step in nine other 4-degree connections (Florida->New Mexico, Florida->Wisconsin, FSU->UTEP, and six involving Kentucky). More connections assuredly exist, but the search function stops at the first permutation it finds (breadth-first, sorted by ESPN's ID field).
Of the 122 other connections between Ole Miss and another FBS school, 65 of them were at 3 degrees of separation.
Note: If someone has some hosting space with PHP/MySQL available, I would be more than happy to provide the script that will ouput a table identical to the one above with the separation between Ole Miss and any other FBS school.