The Resistance Distance between two vertices $i$ and $j$ of a graph $G$ is the effective resistance between them when a battery is attached across them and each edge is replaced by a unit resistor.
I'm having a hard time coming up with a formal definition of this concept. I tried to give some inductive definition, but it is not really clear to me in what order the induction should proceed. According to Wikipedia the resistance distance $\Omega_{ij}$ between $i$ and $j$ is given by $$\Omega_{ij}=\Gamma_{ii}+\Gamma_{jj}-\Gamma_{ij}-\Gamma_{ji}$$ where $Γ$ is the Moore–Penrose inverse of the Laplacian matrix of $G$.
But i dont want to define it this way, because it seems like this is something that should rather be proved, since, at least to me, this property is not obvious from the physical interpratation.
So my question is: Can you come up with a formal definition that better resembles the physical interpretation of resistance distance?
For example I would be very happy with some implicit definition of the form: $\Omega$ is the unique function on pairs of vertices satisfying a bunch of Kirchhoff-like properties.
Let $G=(V,E)$. There is exactly one assignment of potentials $U_k$ to vertices and directed currents $I_{kl}$ to edges such that $I_{kl}=U_k-U_l$ for all $\langle k,l\rangle\in E$, $\sum_{l:\langle k,l\rangle\in E}I_{kl}=\delta_{ki}-\delta_{kj}$ for all $k\in V$, and $U_j=0$. (This would have to be proved, but it’s true from physics.) The resistance is $U_i$ (with the interpretation that a unit current flows through the graph from $i$ to $j$ and $R=\frac UI$).