How we cite our quotes: (Chapter.Paragraph)
Quote #1
"Ignatius graduated smart."
"Graduated smart," Ignatius repeated with some pique. "Please define your terms. Exactly what do you mean by 'graduated smart'?" (1.272-273)
What Mrs. Reilly means, as Ignatius knows, is that Ignatius did very well in school but isn't doing so well with the rest of his life. He is smart in terms of knowing who Boethius is, but in every other way (including physical mass) he is kind of dense.
Quote #2
"Mrs. Levy is a brilliant, educated woman. She's taken a correspondence course in psychology." (3.119)
The joke here is that a correspondence course in psychology isn't much of a guarantee of brilliance and education. We learn later that Mrs. Levy didn't pass her correspondence course (they wouldn't even give her an "F").
Quote #3
"I mingle with my peers or no one, and since I have no peers, I mingle with no one." (5.184)
Ignatius is saying he's too smart, and the quality of his mind is too unique, for him to deal with normal people. Though, of course, the truth is he mingles with lots of people—and in fact, his inertia means that at the end of the novel he's on the front page of the paper for mingling with criminals, pornographers, and the dregs of New Orleans society. (Though many of those dregs, the novel suggests, are actually considerably more pleasant than Ignatius.)