  The low energy expansion of Type II superstring amplitudes at genus one is
organized in terms of modular graph functions associated with Feynman graphs of
a conformal scalar field on the torus. In earlier work, surprising identities
between two-loop graphs at all weights, and between higher-loop graphs of
weights four and five were constructed. In the present paper, these results are
generalized in two complementary directions. First, all identities at weight
six and all dihedral identities at weight seven are obtained and proven.
Whenever the Laurent polynomial at the cusp is available, the form of these
identities confirms the pattern by which the vanishing of the Laurent
polynomial governs the full modular identity. Second, the family of modular
graph functions is extended to include all graphs with derivative couplings and
worldsheet fermions. These extended families of modular graph functions are
shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The
eigenvalues for the extended family of dihedral modular graph functions are
calculated analytically for the simplest infinite sub-families and obtained by
Maple for successively more complicated sub-families. The spectrum is shown to
consist solely of eigenvalues $s(s-1)$ for positive integers $s$ bounded by the
weight, with multiplicities which exhibit rich representation-theoretic
patterns.
