Discuss Edward II's relationship with Gaveston.

Edward's relationship with his favorite Gaveston can be described as more than just platonic. There are clear sexual overtones here, as we can see in the following lines from the play's opening scene:

Kiss not my hand; Embrace me, Gaveston, as I do thee.

This is Edward's impatient response to Gaveston kissing his hand according to court etiquette. His insistence on being embraced by his favorite indicates that the relationship between the two men is a good deal more intimate than would normally be the case between a king and one of his loyal subjects.
Yet in the world that Edward and Gaveston inhabit, love and loyalty are closely linked. In act 1, scene 4, when Mortimer asks the king why he loves Gaveston when the whole world hates him so much, Edward makes a revealing statement:

Because he loves me more than all the world.

Gaveston is not just Edward's lover but also a loyal and faithful servant to his king, which is more than can be said for most of Edward's courtiers. Mortimer is one of those courtiers, and like the rest of the court, he is grievously offended by the open displays of affection between Edward and Gaveston.
Whether or not there really is a sexual dimension to this relationship, there can be no doubting its intensity. There is genuine love between the two men, as can be seen in Edward's heading up north with Gaveston in an attempt to save him from vengeful lords who want to kill him. Ultimately, Edward's efforts are futile, but his strength of feeling for Gaveston remains unimpaired, as can be seen in his bitter mourning over Gaveston's death.