Flow Polynomial

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


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].

The flow polynomial C_G^*(u) of a planar graph G is related to the chromatic polynomial of its dual graph G^* by


The flow polynomial of a bridged graph, and therefore also of a tree on >=2 nodes, is 0.

The flow polynomials for some special classes of graphs are summarized in the table below.

Linear recurrences for some special classes of graphs are summarized below.

See also

Chromatic Polynomial, Rank Polynomial, Tutte Polynomial

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Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 110-111, 1993.Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications I: The Tutte Polynomial." 28 Jun 2008., C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, p. 370, 2001.

Referenced on Wolfram|Alpha

Flow Polynomial

Cite this as:

Weisstein, Eric W. "Flow Polynomial." From MathWorld--A Wolfram Web Resource.

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