  We consider an Sp(4,Z) invariant expression involving two factors of the
Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link,
and four derivatives on the moduli space of genus two Riemann surfaces.
Manipulating it, we show that the integral over moduli space of a linear
combination of a modular graph with two links and the square of the KZ
invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant
expression involving three factors of the KZ invariant and six derivatives on
moduli space, from which we deduce that the integral over moduli space of a
modular graph with three links reduces to a boundary integral. In both cases,
the boundary term is completely determined by the KZ invariant. We show that
both the integrals vanish.

High Energy Physics - Theory;Number Theory