Which was the hegemonic state in Europe during the Dark Ages (476 to 800 AD)?

The hegemonic empire in Europe in the Dark Ages (between the fall of the Western Roman Empire 476 to the coronation of Charlemagne in 800 AD) was the Eastern Roman Empire, also known as the Byzantine Empire, whose capital was Constantinople.

When the Unified Roman Empire split in two after the death of Emperor Theodosius in 395 AD, his older son Arcadius got the Eastern part of the Empire, whose capital was Constantinopolis (Constantinople) and the younger son Honorius got the Western part of the empire, whose capital initially was Mediolanum and later Ravenna.

The Eastern Roman Empire survived the collapse of the Western Empire in 476 AD, and Emperor Justinian went on reconquest. He reconquered much of the territories of the old Western Roman Empire - including the city of Rome itself - and incorporated them back into the Eastern Empire. But the outbreak of plague in 541-542 put the end to the reconquest.

Nevertheless, the Roman culture and civilization survived and thrived in the Eastern Roman Empire, and it was perhaps the strongest empire in the world during the Dark Ages, with the possible exception of the Tang dynasty China.

Note that the name "Byzantine Empire" is a misnomer. It was historically never called this way - it was called simply "Roman Empire", "Romania" or "Eastern Empire".