({{, (RefLink[FlowPolynomial, paclet:ref/FlowPolynomial])[g,k] 
gives the flow polynomial of the graph g.}, {, (RefLink[FlowPolynomial, paclet:ref/FlowPolynomial])[{v->w,…},…]
uses rules v->w to specify the graph g. }})

OpenersNotesSection(Spacer1)Details

(RefLink[FlowPolynomial, paclet:ref/FlowPolynomial])[g,k] gives the number of nowhere-zero k-flows of g.

(RefLink[FlowPolynomial, paclet:ref/FlowPolynomial])[g] gives a pure function representation of the flow polynomial of g.

OpenersPrimaryExamplesSection(Spacer1)Examples  (11)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

The flow polynomial of a Wheel graph:

FlowPolynomial[WheelGraph[3],k]

Plot the polynomial:

DiscretePlot[%,{k,40}]

OpenersExampleSection1(Spacer1)Scope  (6)

(RefLink[FlowPolynomial, paclet:ref/FlowPolynomial]) works with undirected graphs:

FlowPolynomial[,k]

Directed graphs:

FlowPolynomial[,k]

Multigraphs:

FlowPolynomial[,k]

Mixed graphs:

FlowPolynomial[,k]

Use rules to specify the graph:

FlowPolynomial[{1->2,2->5,3->5,4->1,4->3,5->1},k]

Evaluate at specific value:

FlowPolynomial[,4]

OpenersExampleSection3(Spacer1)Applications  (1)

Flow polynomials for wheel graphs with n vertices:

Table[FlowPolynomial[WheelGraph[n],k],{n,3,8}]

FindSequenceFunction[%,n-2]

Circulant graphs with n vertices and two jumps:

Table[FlowPolynomial[CirculantGraph[n,2],k],{n,5,15}]

FindSequenceFunction[%,n-4]

OpenersExampleSection4(Spacer1)Properties & Relations  (3)

Use (RefLink[TuttePolynomial, paclet:ref/TuttePolynomial]) to compute (RefLink[FlowPolynomial, paclet:ref/FlowPolynomial]):

g=WheelGraph[5];

{n,m,c}={VertexCount[g],EdgeCount[g],Length[ConnectedComponents[g]]};

(((-1))^(m-n+c))TuttePolynomial[g,{0,1-k}]//Expand

FlowPolynomial[g,x]

Isomorphic graphs have the same flow polynomial:

g=;h=;

IsomorphicGraphQ[g,h]

FlowPolynomial[g,k]==FlowPolynomial[h,k]

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

FlowPolynomial[CycleGraph[5],k]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[TuttePolynomial, paclet:ref/TuttePolynomial])  ▪  (RefLink[ChromaticPolynomial, paclet:ref/ChromaticPolynomial])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Computation on Graphs, paclet:guide/ComputationOnGraphs])}})}})