You have enough to worry about memorizing without keeping all these terms in your head. We intend to provide this list for your reference during every in-class evaluation relating to graphs.

Term	Definition
Walk	An alternating sequence of vertices and edges starting and ending with a vertex, each edge (x,y) in the walk follows vertex x and is followed by vertex y
Path	A walk that does not visit any vertex twice
Closed Walk	A walk that begins and ends at the same vertex
Cycle	A closed walk that is a path except for its last vertex

The related definitions on relations R : A \rightarrow A are

Term	Definition
R is Reflexive	\forall x \in A \;.\; x R x
R is Irreflexive	\forall x \in A \;.\; \lnot(x R x)
R is Symmetric	\forall x,y \in A \;.\; (x R y) \rightarrow (y R x)
R is Asymmetric	\forall x,y \in A \;.\; (x R y) \rightarrow \lnot(y R x)
R is Antisymmetric	\forall x \neq y \in A \;.\; (x R y) \rightarrow \lnot(y R x)
R is Transitive	\forall x,y,z \in A \;.\; (x R y) \land (y R z) \rightarrow (x R z)

And those lead to these terms:

Term	Definition
Strict partial order	transitive and asymmetric
Weak partial order	transitive, reflexive, and antisymmetric
Equivalence relation	transitive, reflexive, and symmetric