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FlowPolynomial[g,k]

gives the flow polynomial of the graph g.

FlowPolynomial[{vw,…},…]

uses rules vw to specify the graph g.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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FlowPolynomial

FlowPolynomial[g,k]

gives the flow polynomial of the graph g.

FlowPolynomial[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • FlowPolynomial[g,k] gives the number of nowhere-zero k-flows of g.
  • FlowPolynomial[g] gives a pure function representation of the flow polynomial of g.

Examples

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Basic Examples  (1)

The flow polynomial of a Wheel graph:

Plot the polynomial:

Scope  (6)

FlowPolynomial works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Evaluate at specific value:

Applications  (1)

Flow polynomials for wheel graphs with vertices:

Circulant graphs with vertices and two jumps:

Properties & Relations  (3)

Use TuttePolynomial to compute FlowPolynomial:

Isomorphic graphs have the same flow polynomial:

The flow polynomial for a cycle graph is k-1:

Wolfram Research (2014), FlowPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/FlowPolynomial.html (updated 2015).

Text

Wolfram Research (2014), FlowPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/FlowPolynomial.html (updated 2015).

CMS

Wolfram Language. 2014. "FlowPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/FlowPolynomial.html.

APA

Wolfram Language. (2014). FlowPolynomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FlowPolynomial.html

BibTeX

@misc{reference.wolfram_2025_flowpolynomial, author="Wolfram Research", title="{FlowPolynomial}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FlowPolynomial.html}", note=[Accessed: 08-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_flowpolynomial, organization={Wolfram Research}, title={FlowPolynomial}, year={2015}, url={https://reference.wolfram.com/language/ref/FlowPolynomial.html}, note=[Accessed: 08-August-2025]}