  In this paper, we study some vanishing identities for Gromov-Witten
invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in
the case that the summation range is large compare to genus. In fact, in such
cases, we can obtain a vanishing identity which is stronger than their
conjectures. Moreover we will also prove their conjectures in low genus cases.
