We have a simple resistor network \$g\$ with \$5\$ nodes and \$6\$ edges (resistors) as given below:
We assume all resistors have the same resistance (say \$1\$ Ohm, so homogeneously distributed resistances). Now suppose we want to know which edges would carry a non-zero current if we applied a \$1\$ Volt potential difference across the nodes \$1\$ and \$4.\$ Once we apply this constant potential difference, there will be current flowing either from node \$1\$ towards \$4\$ or the other way around. Assuming so far I haven't made a mistake in my description (from circuit theory point of view), if we adopt a "positive" direction for the current (I guess it doesn't matter which way is chosen), how do we determine the current direction along each edge? In other words, having now induced a current flow through the resistor network, my originally undirected graph will become a directed one, but I don't know how the edge directionality ought to be defined consistently.
One hypothetical situation of current flow once a voltage is applied between nodes \$1\$ to \$4\$ could be: (intuitively I don't expect there to be current flowing between 4 -> 5 because probably the potential difference between those nodes is 0 given the difference was applied across 1 and 4.)
Once I can determine the direction of flow across each edge then I can write down the incidence matrix \$A\$ where then for an edge that goes from node \$x\$ to node \$y\$, the row corresponding to that edge has \$−1\$ in column \$x\$ and \$1\$ in column \$y\$ with all other entries in that row being \$0.\$ Having the matrix \$A,\$ to answer my original question (namely which edges are carrying non-zero current), for the current vector \$\mathbf{i}\$, I invoke the Kirchhoff's law, namely that:
$$A^T \mathbf{i} = \mathbf{0},$$
where the current vector has a dimension equal to number of edges in \$g,\$ then finding an eigen-basis for the nullspace of \$A^T,\$ I would know which entries of \$\mathbf{i}\$ are non-zero. I admit I am not entirely sure about this approach, as in whether it's really the way to find an answer to my original question.
Summary of questions:
How do I determine the direction of edges, in other words direction of current flow along each resistor, as described in the first paragraph?
Given the original question of: "which edges will carry a non-zero current once a potential difference is applied across nodes \$1\$ and \$4,\$," is my approach as described in the last paragraph at all sound?
I've specifically chosen a very small and relatively simple network in this question for the purpose of illustration.