# 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 TutteWhitney polynomial.
• 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

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## Basic Examples(1)

The Tutte polynomial of a cycle graph:

Plot the contours of the polynomial:

## Scope(6)

TuttePolynomial works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Evaluate at specific values:

## Applications(6)

Find the number of spanning trees of a complete graph:

Cycle graphs:

Wheel graphs:

Find the number of forests of a cycle graph:

Find the number of spanning subgraphs of a cycle graph:

Find the number of acyclic orientations of a cycle graph:

Find the number of strongly connected orientations of a cycle graph:

Compute graph invariant polynomials:

Chromatic polynomials:

Flow polynomials:

Reliability polynomials:

## Properties & Relations(4)

Isomorphic graphs have the same Tutte polynomial:

The Tutte polynomial of a tree with edges is :

TuttePolynomial[g,{1,1}] counts the number of spanning trees in the graph:

TuttePolynomial[g,{2,2}] is equal to 2EdgeCount[g]: