  To a simple graph we associate a so-called graph series, which can be viewed
as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study
new $q$-representations and examine modular properties of several examples
including Dynkin diagrams of finite and affine type. Notably, we obtain new
formulas for graph series of type $A_7$ and $A_8$ in terms of "sum of tails"
series, and of type $D_4$ and $D_5$ in the form of indefinite theta functions
of signature $(1,1)$. We also study examples related to sums of powers of
divisors corresponding to $5$-cycles. For several examples of graphs, we prove
that graph series are so-called mixed quantum modular forms.

Number Theory;Quantum Algebra