Here we deal with the question of definability of infinite graphs up
to isomorphism by weak monadic second-order formulae. In this
respect, we prove that the quantifier alternation hierarchy of
equational graphs is infinite. Two proofs are given: the first one is
based on the Ehrenfeucht-Fraissé games; the second one uses the
arithmetical hierarchy. Finally we give a new proof of the Thomas's
result according to which the hierarchy of the weak monadic
second-order logic of the complete binary tree is infinite.