Let C^*(u) denote the number of nowhere-zero u-flows on a connected graph G with vertex count n, edge count m, and connected component count c. This quantity is called the flow polynomial of the graph G, and is given by C^*(u) | = | (-1)^m R(-1, - u) | = | (-1)^(m - n + c) T(0, 1 - u), where R(x, y) is the rank polynomial and T(x, y) is the Tutte polynomial (extending Biggs 1993, p. 110). The flow polynomial of a graph g can be computed in the Wolfram Language using FlowPolynomial[g, u].

chromatic polynomial | rank polynomial | Tutte polynomial