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
open all close allBasic Examples (1)
Scope (6)
FlowPolynomial works with undirected graphs:
FlowPolynomial[[image], k]FlowPolynomial[[image], k]FlowPolynomial[[image], k]FlowPolynomial[[image], k]Use rules to specify the graph:
FlowPolynomial[{1 -> 2, 2 -> 5, 3 -> 5, 4 -> 1, 4 -> 3, 5 -> 1}, k]FlowPolynomial[[image], 4]Applications (1)
Flow polynomials for wheel graphs with 
vertices:
Table[FlowPolynomial[WheelGraph[n], k], {n, 3, 8}]FindSequenceFunction[%, n - 2]Circulant graphs with 
vertices and two jumps:
Table[FlowPolynomial[CirculantGraph[n, 2], k], {n, 5, 15}]FindSequenceFunction[%, n - 4]Properties & Relations (3)
Use TuttePolynomial to compute FlowPolynomial:
g = WheelGraph[5];{n, m, c} = {VertexCount[g], EdgeCount[g], Length[ConnectedComponents[g]]};(-1)^m - n + cTuttePolynomial[g, {0, 1 - k}]//ExpandFlowPolynomial[g, x]Isomorphic graphs have the same flow polynomial:
g = [image];h = [image];IsomorphicGraphQ[g, h]FlowPolynomial[g, k] == FlowPolynomial[h, k]The flow polynomial for a cycle graph is k-1:
FlowPolynomial[CycleGraph[5], k]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