  In this paper we study the algebra of graph invariants, focusing mainly on
the invariants of simple graphs.
  All other invariants, such as sorted eigenvalues, degree sequences and
canonical permutations, belong to this algebra. In fact, every graph invariant
is a linear combination of the basic graph invariants which we study in this
paper.
  To prove that two graphs are isomorphic, a number of basic invariants are
required, which are called separator invariants. The minimal set of separator
invariants is also the minimal basic generator set for the algebra of graph
invariants.
  We find lower and upper bounds for the minimal number of generator/separator
invariants needed for proving graph isomorphism.
  Finally we find a sufficient condition for Ulam's conjecture to be true based
on Redfield's enumeration formula.
