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 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.

Nerva had rumors recorded about him having affairs with men, and was a lifelong bachelor which was EXTREMELY unusual even for Romans who preferred men.

Now, Trajan? He was well-known for preferring male company.

Trajan was most conspicuous for his justice, for his bravery, and for the simplicity of his habits. He was strong in body, being in his forty-second year when he began to rule, so that in every enterprise he toiled almost as much as the others; and his mental powers were at their highest, so that he had neither the recklessness of youth nor the sluggishness of old age. He did not envy nor slay any one, but honoured and exalted all good men without exception, and hence he neither feared nor hated any one of them. To slanders he paid very little heed and he was no slave of anger. He refrained equally from the money of others and from unjust murders. He expended vast sums on wars and vast sums on works of peace; and while making very many urgently needed repairs to roads and harbours and public buildings, he drained no one’s blood for any of these undertakings. He was so high-minded and generous that, after enlarging and embellishing the Circus, which had crumbled away in places, he merely inscribed on it a statement that he had made it adequate for the Roman people.

For these deeds, now, he took more pleasure in being loved than in being honoured. His association with the people was marked by affability and his intercourse with the senate by dignity, so that he was loved by all and dreaded by none save the enemy. He joined others in the chase and in banquets, as well as in their labours and plans and jests. Often he would take three others into his carriage, and he would enter the houses of citizens, sometimes even without a guard, and enjoy himself there. Education in the strict sense he lacked, when it came to speaking, but its substance he both knew and applied; and there was no quality which he did not possess in a high degree.

I know, of course, that he was devoted to boys and to wine, but if he had ever committed or endured any base or wicked deed as the result of this, he would have incurred censure; as it was, however, he drank all the wine he wanted, yet remained sober, and in his relation with boys he harmed no one. And even if he did delight in war, nevertheless he was satisfied when success had been achieved, a most bitter foe overthrown and his countrymen exalted. Nor did the result which usually occurs in such circumstances — conceit and arrogance on the part of the soldiers — ever manifest itself during his reign; with such a firm hand did he rule them.

• Cassius Dio