Explanation: The three childless gay Emperors are Nerva, Trajan, and Hadrian.
Nerva was a temperate and even-keeled man elected by the Senate after Domitian was overthrown, and was a lifelong bachelor with rumors that he had love affairs with men (male affairs were not all that exceptional for Roman men, just something that was noted by the notoriously gossippy Roman histories). He adopted Trajan, who was an adult already and a popular military man.
Trajan was married and had a cordial relationship with his wife, but was quite openly and famously attracted to men. He was, as mentioned, a military man, and one who brought the Roman Empire to its greatest extent, but was noted for having a spirit of cooperation, civility, and due process rather than the dictatorial military mien one might expect from a career soldier. For this reason, the Senate adored him and called him ‘Optimus Princeps’ - ‘The Best Emperor’, a title later historians would keep for him.
Hadrian, Trajan’s adopted son, was also militaristic and quite openly gay. Unlike Trajan, Hadrian had a bit of a temper and a contentious relationship with the Senate - and his wife! Hadrian, however, was also a cultured and dutiful Emperor who attempted extensive legal reforms to establish the rule of law over the pre-eminence of the Emperor. Mostly it didn’t outlast him, but it was a nice thought! He, quite famously, deified his (male) lover Antinous, and Antinous would remain a symbol of male-male sexual attraction for the next ~1800 years.
Hadrian adopted Antoninus Pius, who was straight - in fact, Pius is one of a very small number of Roman Emperors we have no rumors of same-sex activity about. He was noted for his mildness, his justice, and love of peace. Pius never remarried after the death of his first wife, having loved her deeply, and had no biological sons.
Pius adopted Marcus Aurelius and Lucius Verus - Lucius Verus died fairly young in his Emperorship.
Marcus Aurelius had a biological son, Commodus, who was… not a good Emperor. At all.