Graph schemes, graph series, and modularity

To a simple graph we associate so-called series, which can be viewed as the Hilbert–Poincaré series of certain infinite jet scheme. We study new q-representations and examine modular properties several examples including Dynkin diagrams finite affine type. Notably, obtain formulas for type A7 A8 in terms “sum tails” D4 D5 form indefinite theta functions signature (1,1). also related to sums powers divisors corresponding 5-cycles. For graphs, prove that are mixed quantum forms mock forms.