  The expansion of a modular graph function on a torus of modulus $\tau$ near
the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with
coefficients that are rational multiples of single-valued multiple zeta-values,
apart from the leading term whose coefficient is rational and exponentially
suppressed terms. We prove that the coefficients of the non-leading terms in
the Laurent polynomial of the modular graph function $D_N(\tau)$ associated
with a melon graph is free of irreducible multiple zeta-values and can be
written as a polynomial in odd zeta-values with rational coefficients for
arbitrary $N \geq 0$. The proof proceeds by expressing a generating function
for $D_N(\tau)$ in terms of an integral over the Virasoro-Shapiro closed-string
tree amplitude.
