Formal definition

The transitive_objects method finds all nodes such that there is a path from subject to one of those nodes using only the predicate property in the triples. The transitive_subjects method is similar; it finds all nodes such that there is a path from the node to the object using only the predicate property.

Informal description, with an example

In brief, transitive_objects walks forward in a graph using a particular property, and transitive_subjects walks backward. A good example uses a property ex:parent, the semantics of which are biological parentage. The transitive_objects method would get all the ancestors of a particular person (all nodes such that there is a parent path between the person and the object). The transitive_subjects method would get all the descendants of a particular person (all nodes such that there is a parent path between the node and the person). So, say that your URI is ex:person. The following code would get all of your (known) ancestors, and then get all the (known) descendants of your maternal grandmother:

from rdflib import ConjunctiveGraph, URIRef
 
person = URIRef('ex:person')
dad = URIRef('ex:d')
mom = URIRef('ex:m')
momOfDad = URIRef('ex:gm0')
momOfMom = URIRef('ex:gm1')
dadOfDad = URIRef('ex:gf0')
dadOfMom = URIRef('ex:gf1')
 
parent = URIRef('ex:parent')
 
g = ConjunctiveGraph()
g.add((person, parent, dad))
g.add((person, parent, mom))
g.add((dad, parent, momOfDad))
g.add((dad, parent, dadOfDad))
g.add((mom, parent, momOfMom))
g.add((mom, parent, dadOfMom))
 
print "Parents, forward from `ex:person`:"
for i in g.transitive_objects(person, parent):
  print i
 
print "Parents, *backward* from `ex:gm1`:"
for i in g.transitive_subjects(parent, momOfMom):
  print i

Warning: The transitive_objects method has the start node as the first argument, but the transivite_subjects method has the start node as the second argument.