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Tutte polynomial

Tutte polynomial is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tutte polynomial rather than just read about it. In short: The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory.

Tutte polynomial — main illustration
Tutte polynomial — illustration

Key takeaways

  • Tutte polynomial belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Tutte polynomial to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Tutte polynomial from memory before moving on to harder problems.

Reference excerpt

The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G {\displaystyle G} and contains information about how the graph is connected. It is denoted by T G {\displaystyle T_{G}} . The importance of this polynomial stems from the information it contains about G {\displaystyle G} . Though originally studied in algebraic graph theory as a generalization of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specializations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science. The Tutte polynomial has several equivalent definitions. It is essentially equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin–Kasteleyn’s random cluster model under simple transformations. It is essentially a generating function for the number of edge sets of a given size and number of connected components, with immediate generalizations to matroids. It is also the most general graph invariant that can be defined by a deletion–contraction recurrence. Several textbooks about graph theory and matroid theory devote entire chapters to it.

Definitions Definition. For an undirected graph G = ( V , E ) {\displaystyle G=(V,E)} one may define the Tutte polynomial as

T G ( x , y ) = ∑ A ⊆ E ( x − 1 ) k ( A ) − k ( E ) ( y − 1 ) k ( A ) + | A | − | V | , {\displaystyle T_{G}(x,y)=\sum \nolimits _{A\subseteq E}(x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|},}

where k ( A ) {\displaystyle k(A)} denotes the number of connected components of the graph ( V , A ) {\displaystyle (V,A)} . In this definition it is clear that T G {\displaystyle T_{G}} is well-defined and a polynomial in x {\displaystyle x} and y {\displaystyle y} . The same definition can be given using slightly different notation by letting r ( A ) = | V | − k ( A ) {\displaystyle r(A)=|V|-k(A)} denote the rank of the graph ( V , A ) {\displaystyle (V,A)} . Then the Whitney rank generating function is defined as

R G ( u , v ) = ∑ A ⊆ E u r ( E ) − r ( A ) v | A | − r ( A ) . {\displaystyle R_{G}(u,v)=\sum \nolimits _{A\subseteq E}u^{r(E)-r(A)}v^{|A|-r(A)}.}

The two functions are equivalent under a simple change of variables:

T G ( x , y ) = R G ( x − 1 , y − 1 ) . {\displaystyle T_{G}(x,y)=R_{G}(x-1,y-1).}

Tutte’s dichromatic polynomial Q G {\displaystyle Q_{G}} is the result of another simple transformation:

T G ( x , y ) = ( x − 1 ) − k ( G ) Q G ( x − 1 , y − 1 ) . {\displaystyle T_{G}(x,y)=(x-1)^{-k(G)}Q_{G}(x-1,y-1).}

Tutte’s original definition of T G {\displaystyle T_{G}} is equivalent but less easily stated. For connected G {\displaystyle G} we set

… excerpt ends here. Continue reading the full article.

Illustrations

Tutte polynomial: The polynomial x 4 + x 3 + x 2 y {\displaystyle x^{4}+x^{3}+x^{2}y} is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane y = 0 {\displaystyle y=0} , which is essentially equivalent to the chromatic polynomial.
The polynomial x 4 + x 3 + x 2 y {\displaystyle x^{4}+x^{3}+x^{2}y} is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane y = 0 {\displaystyle y=0} , which is essentially equivalent to the chromatic polynomial.
Tutte polynomial: Example calculation of Tutte polynomial via deletion-contraction formula.
Example calculation of Tutte polynomial via deletion-contraction formula.
Tutte polynomial: Example calculation of Tutte polynomial by enumeration of spanning trees, with indication of internal and external activities in red and blue respectively.
Example calculation of Tutte polynomial by enumeration of spanning trees, with indication of internal and external activities in red and blue respectively.
Tutte polynomial: The chromatic polynomial drawn in the Tutte plane
The chromatic polynomial drawn in the Tutte plane
Tutte polynomial: The Jones polynomial drawn in the Tutte plane
The Jones polynomial drawn in the Tutte plane

Worked examples

Example 1 — a first encounter with Tutte polynomial

Start with the simplest possible case. Write down what Tutte polynomial claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tutte polynomial before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tutte polynomial ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tutte polynomial

In research
Tutte polynomial appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Tutte polynomial in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Tutte polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems, Duality (mathematics), Graph invariants, so understanding it makes those chapters shorter.
In everyday life
Look for Tutte polynomial outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Tutte polynomial in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Tutte polynomial means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Tutte polynomial out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tutte polynomial in simple terms?

The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory.

Why does Tutte polynomial matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tutte polynomial?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tutte polynomial.

Tags

  • Computational problems
  • Duality (mathematics)
  • Graph invariants
  • Matroid theory
  • Polynomials

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