  The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the
equivalence of three distinguishing properties of prime ideals in a
non-commutative algebra A. These properties are of (i) being primitive, (ii)
being rational, and (iii) being locally closed in the Zariski topology of
Spec(A). The DM-equivalence holds in many interesting algebras over a field.
Recently, it was shown that the prime ideals of a Leavitt path algebra of a
finite graph satisfy the DM-equivalence, In this note, we investigate the
occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary
directed graph E. Our analysis shows that locally closed prime ideals of L
satisfy interesting equivalent properties such as being strongly primitive and
completely irreducible. Examples illustrate the results and the constraints.
