If you suspect that the end cap as been removed, not by sound, or smell, but by looking at it or feeling it, then you can suspect the bat has been shaved.
Rolled is trickier. A rolled bat, in theory, isn't supposed to perform beyond 1.15 BPF, as that's exactly what the testers do, until the bat fails. Most folks will take a composite bat to the cages, or hit a ton of soft toss to break in a bat. Others will try to shortcut the process, and get it rolled. Some, maybe most, will condsider rolling to be altering, and thus illegal.
If you consider rolling to be tampering, and feel those long edges along the barrel, like some crude rollers leave, feel free to toss it. Good rollers won't leave that evidence, though. Nor will a "good" shaver. They can replace an endcap, and leave no evidence behind. There's a market for tampered bats, mostly by adult softball goons, but that's trickled down to youth ball, by idiot Dads. Sad, but true. And fellas, you can't tell, if a pro did it.
But if you can't tell for sure, there's nothing you can do about it. Weighing it won't prove anything, as I'm sure most bats don't come from the factory in the exact weight stated on the barrel. Plus, pro shavers will add weight to make up for the stuff taken out of the barrel.
What Presidents CAN do is warn their managers that their jobs are on the line if a tampered bat enters a game, since they appoint them.
Carbon bats have a certain life cycle, and take on different characteristics throughout. They start out stiff, and lifeless. As the fibers get broken down, the trampolin effect starts to happen, they get more lively. You'll even hear a difference, and may even hear loose fibers rolling around inside some of the carbon lined bats. That's normal. Lastly, they'll either go mushy and dead, or crack.
Okay Mike, so what's going on in your league?