Naturality as an affine automorphism action

Definitions and mathematical statement

Let $G=(V,E)$ be a finite connected loopless undirected graph, allowing parallel edges. Its Laplacian acts on $f\in\mathbb Z^V$ by $(\Delta f)(v)=\sum_{vw\in E}(f(v)-f(w))$, counting edge multiplicity. Let $\operatorname{Div}^0(G)=\{a\in\mathbb Z^V:\sum_v a(v)=0\}$ and define the finite critical group \[ \operatorname{Jac}(G)=\operatorname{Div}^0(G)/\Delta\mathbb Z^V. \] Let $\mathcal T(G)$ be the set of spanning trees, with parallel edges treated as distinct choices. An action of $\operatorname{Jac}(G)$ on $\mathcal T(G)$ is simply transitive if, for every $T,T'$, there is exactly one $a\in\operatorname{Jac}(G)$ with $a\cdot T=T'$.

Determine for which graphs there exists such an action that is equivariant under every graph automorphism: \[ \sigma(a\cdot T)=\sigma_*(a)\cdot\sigma(T) \quad(\sigma\in\operatorname{Aut}(G)). \] Here $\sigma_*$ is the automorphism induced by permuting divisor coordinates, and $\sigma(T)$ is the image tree, including the induced edge permutation. This states precisely the minimum naturality requirement in the workshop. The matrix-tree equality of the two cardinalities guarantees that arbitrary simply transitive actions can be transported along bijections, but such choices need not respect automorphisms. The desired action should require neither a distinguished sink nor a cyclic ordering of edges around vertices. The target is the existence and characterization of equivariant torsor structures, not merely a bijection between two finite sets of equal size.

Short English statement

Which graphs admit an automorphism-compatible, simply transitive critical-group action on their spanning trees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000973 (AIM-COMBINATORICS-0098), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Generalizations of chip-firing and the critical group, item 8. Original workshop question underlying AIM-COMBINATORICS-0098. https://aimath.org/pastworkshops/chipfiringproblems.pdf.