In the novel of Lord of the Flies, how would the book's events be different if the characters were comprised of one grown-up and the rest children?

I believe that by "children" you mean that the characters are exactly the same as before.  The only difference is that there is now a single adult among the group.  
This is a completely open question.  It's up to you to explain what you think and why.  
For me, the question is almost too wide open to give a prediction.  What kind of adult are we talking about?  If the adult is a natural leader that happens to be a male, then I think that he steps into the leadership role and all of the boys follow his lead.  I do not think that the chaos ensues that Golding wrote about, because I don't feel that characters like Jack would have the brazen confidence to disregard social norms.  The adult is a feared authority figure.  
On the other hand, if the adult is a female, I feel that the chaos might still happen.  I have nothing against women, but it definitely takes a strong lady to control a group comprised of all teenage boys.  I had a colleague one year that had an eighth grade history class made up of 23 boys.  There were no girls.  This colleague of mine was a very good teacher, and well respected across multiple classes and ages; however, she would frequently comment on how her all boy history class was the toughest class that she ever had to teach. An adult female on the island might have very little impact on the events of the story.