Representing first order sentences as conceptual graphs
Solution 1:
I don't really know about the technical details of Conceptual Graphs, but I do know that it is based off Existential Graphs, and here are the respective representations in Existential Graphs:
$\forall x \ part(x,x)$:
$\forall x \forall y \ ((part(x,y) \land part(y,x)) \to x = y)$:
$\forall x \forall y \forall z \ ((part(x,y) \land part(y,z)) \to part(x,z))$:
The last two certainly look like what you have!