Usually we've considered the games graph to be defined as teams A and B are "connected" if they play each other, or each played a team that the other played. But we could also treat the games graph as a directed graph. Instead of A↔B meaning teams A and B played, we only note A→B when A wins a game vs B.
We can't talk about the "connectivity" of the directed graph, because obviously any undefeated team cannot be "connected to" by any other team, and a winless team does not "connect to" any other team. There is an equivalent to the "diameter" of the games graph, though, and it's not "the longest path" between any two teams.
Define the diameter of the graph as "the largest number of steps required to connect every team to be all the teams that it can be connected to by a victory chain." Note that this will be lower than the longest chain, since step N can add chains longer than N.
In general if there's an A→…→Z chain there's also a Z→…→A chain. You can always find a victory chain connecting two teams except:
2006-10-28 @ Temple 28 Bowling Green 14 2006-10-14 @ Bowling Green 24 E Michigan 21 2006-10-21 @ E Michigan 17 Toledo 13 2006-09-15 @ Toledo 37 Kansas 31 2006-10-28 @ Kansas 20 Colorado 15 2006-10-14 @ Colorado 30 Texas Tech 6 2006-11-04 @ Texas Tech 55 Baylor 21 2006-09-30 @ Baylor 17 Kansas St 3 2006-11-11 @ Kansas St 45 Texas 42 2006-10-07 Texas 28 Oklahoma 10 2006-09-09 @ Oklahoma 37 Washington 20 2006-09-23 @ Washington 29 UCLA 19 2006-11-11 @ UCLA 25 Oregon St 7 2006-10-28 @ Oregon St 33 USC 31 2006-09-02 USC 50 @ Arkansas 14 Transitivity proves that Temple is better than Arkansas. This 15 game conquering path predicts: Temple over Arkansas by 208 points.
2006-09-16 Arkansas 21 @ Vanderbilt 19 2006-09-30 @ Vanderbilt 43 Temple 14 Transitivity proves that Arkansas is better than Temple. This 2 game conquering path predicts: Arkansas over Temple by 31 points.
If you could look at every such chain (there were 42,523 of them last time I looked) you could come up with a way to rank all 119 teams, just as the computer ratings do (they are better at making thousands of comparisons than we are.)
First, you'd want to a way to choose between chains like Temple→Vanderbilt and Vanderbilt→Temple, so we can say A→…→Z is stronger than Z→…→A if:
That cuts the number of chains down considerably, but you could still have more than 10,000 to turn into a ranking. The next step is to find a way to summarize all that data by team.
Current values for Division 1-A is an index to details for the win chains from every team to every other team.
The "current values" table is ordered by WeightedWins, which combines the number and length of the shortest win chains between teams. For a pair of teams A and B, if the win path from A to B is shorter than or equal to that from B to A, the contribution of B to A's weighted wins is:
So if A's shortest win path to B is 2, each such path is worth 1/2, if it's A→C→D→B each is worth 1/4, and so on. Naturally, a head to head win is worth 1.