Let denote the number of nowherezero flows on a connected graph with vertex count , edge count , and connected component count . This quantity is called the flow polynomial of the graph , and is given by
(1)
 
(2)

where is the rank polynomial and is the Tutte polynomial (extending Biggs 1993, p. 110).
The flow polynomial of a graph can be computed in the Wolfram Language using FlowPolynomial[g, u].
The flow polynomial of a planar graph is related to the chromatic polynomial of its dual graph by
(3)

The flow polynomial of a bridged graph, and therefore also of a tree on nodes, is 0.
The flow polynomials for some special classes of graphs are summarized in the table below.
graph  flow polynomial 
book graph  
cycle graph  
ladder graph  
prism graph  
web graph  0 
wheel graph 
Linear recurrences for some special classes of graphs are summarized below.
graph  order  recurrence 
antiprism graph  4  
book graph  2  
ladder graph  1  
prism graph  3  
wheel graph  2 