TuttePolynomial
TuttePolynomial[g,{x,y}]
gives the Tutte polynomial of the graph g.
TuttePolynomial[{vw,…},…]
uses rules vw to specify the graph g.
Details
- TuttePolynomial is also known as dichromate polynomial or Tutte–Whitney polynomial.
- TuttePolynomial[g] gives a pure function representation of the Tutte polynomial of g.
- For an undirected graph
with 
vertices and 
connected components, the Tutte polynomial is defined as the sum of 
over all subsets 
of edges of 
. 
is the number of connected components of the graph generated by 
with 
vertices.
Examples
open all close allBasic Examples (1)
Scope (6)
TuttePolynomial works with undirected graphs:
TuttePolynomial[[image], {x, y}]TuttePolynomial[[image], {x, y}]TuttePolynomial[[image], {x, y}]TuttePolynomial[[image], {x, y}]Use rules to specify the graph:
TuttePolynomial[{1 -> 3, 2 -> 1, 3 -> 6, 4 -> 6, 1 -> 5, 5 -> 4, 6 -> 1}, {x, y}]TuttePolynomial[[image], {1, 1}]Applications (6)
Find the number of spanning trees of a complete graph:
Table[TuttePolynomial[CompleteGraph[n], {1, 1}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Table[TuttePolynomial[CycleGraph[n], {1, 1}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Table[TuttePolynomial[WheelGraph[n], {1, 1}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Find the number of forests of a cycle graph:
Table[TuttePolynomial[CycleGraph[n], {2, 1}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Find the number of spanning subgraphs of a cycle graph:
Table[TuttePolynomial[CycleGraph[n], {1, 2}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Find the number of acyclic orientations of a cycle graph:
Table[TuttePolynomial[CycleGraph[n], {2, 0}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Find the number of strongly connected orientations of a cycle graph:
Table[TuttePolynomial[CycleGraph[n], {0, 2}], {n, 1, 10}]FindSequenceFunction[%, n]Table[%, {n, 1, 13}]Compute graph invariant polynomials:
g = PetersenGraph[]n = VertexCount[g];m = EdgeCount[g];c = Length[ConnectedComponents[g]];(-1)^n - ck^c * TuttePolynomial[g, {1 - k, 0}] //Expand(-1)^m - n + cTuttePolynomial[g, {0, 1 - k}] //Expand(1 - p)^n - cp^m - n + cTuttePolynomial[g, {1, 1 / p}]//ExpandProperties & Relations (4)
Isomorphic graphs have the same Tutte polynomial:
g = [image];h = [image];IsomorphicGraphQ[g, h]TuttePolynomial[g, {x, y}] == TuttePolynomial[h, {x, y}]The Tutte polynomial of a tree with 
edges is 
:
g = KaryTree[10, 4];TreeGraphQ[g]{TuttePolynomial[g, {x, y}], x^EdgeCount[g]}TuttePolynomial[g,{1,1}] counts the number of spanning trees in the graph:
TuttePolynomial[GridGraph[{2, 3}], {1, 1}]TuttePolynomial[g,{2,2}] is equal to 2EdgeCount[g]:
g = GridGraph[{2, 3}];{TuttePolynomial[g, {2, 2}], 2^EdgeCount[g]}Text
Wolfram Research (2014), TuttePolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/TuttePolynomial.html (updated 2015).
CMS
Wolfram Language. 2014. "TuttePolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/TuttePolynomial.html.
APA
Wolfram Language. (2014). TuttePolynomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TuttePolynomial.html