Proving Effective Resistance is Consistent across Edges in an Edge-Transitive Graphs I am trying to prove that in an edge-transitive graph, the effective resistance across every edge is the same. Intuitively, it makes sense that the effective resistance would be consistent across all edges in an edge-transitive graph, but I am struggling to provide a formal proof. Additionally, I think this result may not hold for vertex-transitive graphs in general, but I am unable to find any counter-examples.
 A: In general, if graphs $G$ and $H$ are isomorphic and $f\colon V(G) \to V(G)$ is an isomorphism, then the effective resistance in graph $G$ across edge $uv$ should be equal to the effective resistance in graph $H$ across edge $f(u)f(v)$.
How to prove this depends on your definition of effective resistance; for example, we can prove that $f$ gives us a bijection between spanning trees of $G$ and of $H$, in which spanning trees of $G$ containing $uv$ correspond to spanning trees of $H$ containing $f(u)f(v)$. Ultimately, this ought to be true for any graph-theoretic property of $G$, which effective resistance is.
The statement for edge-transitive graphs is just a special case, where $f$ is an automorphism of $G$ that takes one edge to another.

As for vertex-transitive graphs, you are right that the generalization is false, and we might expect any vertex-transitive graph that is not edge-transitive to give us a counter-example. For instance, take the triangular prism graph: six of the edges here have effective resistance $\frac8{15}$, and the other three have effective resistance $\frac35$.
