Well, I was reading the Wikipedia page 'List of Byzantine emperors' but I'm not convinced by its arguments mostly when I read: " It was under Constantine that the major characteristics of what is considered the Byzantine state emerged: a Roman polity centered at Constantinople and culturally dominated by the Greek East, with Christianity as the state religion."
If Christianity as the state religion is a presuppost to have a Byzantine state then we should start the list with Theodosius. What do you think?
There is no consensus about when 'Roman' becomes inappropriate and when 'Byzantine' becomes more 'accurate'. Historians regularly use Constantine, Justinian, or the seventh century as a turning point, but, as I argued here, I don't feel very strongly about this debate - everyone knows what the word 'Byzantine' refers to, so it's absolutely fine to use it to refer to the empire (others will disagree of course). There is also an ongoing debate on 'Byzantine' identity, on which I mentioned a few articles at the end of my linked answer, which may prove interesting, as it shows clearly just how even academics are not settled on when it is appropriate to use the word 'Byzantine'!