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FlowPolynomial
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FlowPolynomial

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gives the flow polynomial of the graph g.

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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

open allclose all

Basic Examples  (1)Summary of the most common use cases

The flow polynomial of a Wheel graph:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-bgfmjw
Out[1]=1

Plot the polynomial:

In[2]:=2
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https://wolfram.com/xid/0b8cyf93yq-p9t5l7
Out[2]=2

Scope  (6)Survey of the scope of standard use cases

FlowPolynomial works with undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-yt21g
Out[1]=1

Directed graphs:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-itaq3u
Out[1]=1

Multigraphs:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-ffb0ew
Out[1]=1

Mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-czvddh
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-bndh30
Out[1]=1

Evaluate at specific value:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-lbjtot
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Flow polynomials for wheel graphs with vertices:

In[27]:=27
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https://wolfram.com/xid/0b8cyf93yq-bzjsri
Out[27]=27
In[28]:=28
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https://wolfram.com/xid/0b8cyf93yq-e93rrv
Out[28]=28

Circulant graphs with vertices and two jumps:

In[17]:=17
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https://wolfram.com/xid/0b8cyf93yq-ihdjr4
Out[17]=17
In[18]:=18
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https://wolfram.com/xid/0b8cyf93yq-mlxx65
Out[18]=18

Properties & Relations  (3)Properties of the function, and connections to other functions

Use TuttePolynomial to compute FlowPolynomial:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-e14kk3
In[2]:=2
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https://wolfram.com/xid/0b8cyf93yq-h8pt5
In[3]:=3
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https://wolfram.com/xid/0b8cyf93yq-l0k3mc
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b8cyf93yq-eccxor
Out[4]=4

Isomorphic graphs have the same flow polynomial:

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-iji2i
In[2]:=2
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https://wolfram.com/xid/0b8cyf93yq-be3e60
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b8cyf93yq-dig49q
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0b8cyf93yq-grpg65
Out[1]=1
Wolfram Research (2014), FlowPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/FlowPolynomial.html (updated 2015).
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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).

✖
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.

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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

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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: 21-April-2025 ]}

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@misc{reference.wolfram_2025_flowpolynomial, author="Wolfram Research", title="{FlowPolynomial}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FlowPolynomial.html}", note=[Accessed: 21-April-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: 21-April-2025 ]}

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@online{reference.wolfram_2025_flowpolynomial, organization={Wolfram Research}, title={FlowPolynomial}, year={2015}, url={https://reference.wolfram.com/language/ref/FlowPolynomial.html}, note=[Accessed: 21-April-2025 ]}