In Alien: Covenant (2017), David showed Elizabeth's grave to Walter, saying he loved her very much. In reality, he experimented on her and killed her. If David loved Elizabeth so much, why did he kill her?
Two possible explanations:
David is crazy/evil and lies to Walter. Since he wants to get his hands on ship with colonists he doesn't exactly want Walter to realize he is capable of killing people.
Elizabeth was accidentaly infected. Since David is obsessed with his experiments, he studied and kept records of her case.
Case 1 is much more likely, although kind of both at the same time may apply.