Flow ($\theta$)

A flow is an antisymmetric function on directed edges whose divergence of a flow vanishes away from designated sources and sinks.

A unit flow from $a$ to $b$ is an antisymmetric flow with net outward strength of a flow one at $a$, net inward strength one at $b$, and zero divergence of a flow elsewhere. For a unit flow to infinity, only the finite source has nonzero divergence.

The divergence of a flow at a vertex is its total outward flow. Flow conservation means that this divergence vanishes at every interior vertex.

Flow conservation requires total inflow to equal total outflow at every vertex other than the source and sink, equivalently $\mathrm{div}f=0$ there.

The strength of a source-to-sink flow is the net amount leaving the source, equivalently the net amount entering the sink.

For unit edge resistances, the energy of a flow iswhere every unoriented edge is counted once.

An infinite locally finite connected graph is a transient graph exactly when it supports a unit flow from a vertex to infinity with finite energy of a flow.