Current flow ($i(x,y)$)

A current flow is an antisymmetric function on oriented edges satisfying Kirchhoff's node law away from its sources and sinks. For conductance $c(x,y)$ and voltage $u$, Ohm's law gives $i(x,y)=c(x,y)(u(x)-u(y))$.

The Kirchhoff cycle law says the directed sum of voltage drops around every graph cycle is zero. For unit edge resistance this is the sum of signed currents around the graph cycle. With unit edge resistance, the current is a voltage gradient: define the voltage difference by summing drops along a graph path. The difference of the sums along two graph paths is a sum around closed walks; decomposing these into graph cycles makes it zero. Together with the Kirchhoff node law, the resulting graph Laplacian equation determines current uniquely on a finite connected network.

The Kirchhoff node law says the sum of outward currents at a graph vertex is zero unless that graph vertex is a source or sink. In an electrical network, it is flow conservation; a unit flow has divergences $+1$ and $-1$ at its terminals.