In this question I am mainly concerned with phase transitions associated with symmetry breaking. For phase transitions of this type, there are a number of results describing the relationship between the existence of a phase transition and the dimension of the system.
For example, roughly speaking, the Mermin-Wagner theorem states that there cannot be spontaneous breaking of a continuous symmetry at finite temperature in dimensions $d \leq 2$. This has been proven rigorously for a number of models, including for example the quantum spin-$S$ Heisenberg model with isotropic and finite-range interactions on a $d$-dimensional hypercubic lattice (c.f. Scholarpedia).
Broadly I am interested in whether it is possible to obtain results of this type (i.e. relating the existence of a phase transition to the dimension of the system) for models defined on more general graphs than regular lattices. For a graph $G$, one notion of dimension is the graph dimension, which roughly corresponds to the minimum dimension $d$ for which there is an embedding of the graph in $d$-dimensional Euclidean space such that all edges are of unit length.
To make this question more answerable, here is a specific question:
Is there a version of the Mermin-Wagner theorem for the quantum spin-$S$ Heisenberg model defined on a general graph $G$ (say with nearest-neighbour interactions), relating the existence of a phase transition to the graph dimension of $G$?
I would also be interested in answers which provide counterexamples to this idea.