Orders of nut graphs with prescribed vertex and edge orbit counts

Determine, for every r ∈ N and k ≥ r + 1, all graph orders n for which there exists a nut graph of order n with exactly r vertex orbits and k edge orbits.

Background

The paper establishes that for every r ∈ N and k ≥ r + 1, infinitely many nut graphs have r vertex orbits and k edge orbits, thereby resolving existence and infinite realizability at the level of orbit counts. It does not classify the orders of such graphs. The final problem asks for this finer order-by-order realizability characterization.