Use the noun paternity to describe the relationship a father has to his child. You might say, "His paternity is so clear — his daughter has the same red hair that he does!"
Fatherhood and the connection between a father and child is what paternity is all about, but the word most often comes up in legal situations. Attorneys use the word paternity when there is a question about who a child's biological father is or what kind of financial or legal responsibility he has. The Latin root is paternus, "relating to a father," from pater, "father."