After the superscript (The Song of Songs, which is Solomon's [1:1]), Solomon's name is only mentioned twice ("the litter of Solomon" [3:7] and "you, O Solomon, may have the thousand [vineyards]" [8:12]). But despite the lack of name-dropping, many people still assert that King Solomon himself is the groom.
To be honest, we're not so sure.
If Solomon were actually the groom, why mention him only twice in the whole poem? After all, when you write about someone that famous, you name-drop like crazy, right? If you were reporting on Brangelina's wedding, wouldn't you say Brad and Angelina a few times?
Also, Song of Songs sure goes to great lengths to tout this one lady above all others. Solomon, on the other hand, was famous for having hundreds of wives and plenty of concubines. Sure, the text never comes out and says, "polygamy is bad," but we just can't picture Solomon settling down with this lily of the valley.