# 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.
• 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_2024_flowpolynomial, author="Wolfram Research", title="{FlowPolynomial}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FlowPolynomial.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_flowpolynomial, organization={Wolfram Research}, title={FlowPolynomial}, year={2015}, url={https://reference.wolfram.com/language/ref/FlowPolynomial.html}, note=[Accessed: 16-September-2024 ]}