Examples of Tutte Polynomials

We display the polynomials in the form of a matrix. The entry in position (i+1,j+1) is the coefficient of xⁱy^j. We omit trailing zeroes in these matrices. We begin by showing one example of the Tutte Polynomial and of the two versions of the Rank Polynomial previously mentioned.

The cycle of length four. Tutte Polynomial, T(G;x,y):

This represents the polynomial x+x²+x³+y.

Rank Polynomial, R(G;x,y):

Rank Polynomial, R'(G;x,y):

The above example demonstrates (in a small way) that the coefficients of the Tutte Polynomial are smaller than the coefficients of the Rank Polynomial. For these remaining examples, we display only the Tutte Polynomial.

The complete graph on one vertex.

The complete graph on two vertices.

         0
         1

The complete graph on three vertices.

         0    1
         1
         1

The complete graph on four vertices.

         0    2    3    1
         2    4
         3
         1

When the graph is self dual, the matrix is symmetric.

The complete graph on five vertices.

         0    6   15   15   10    4    1
         6   20   15    5
        11   10
         6
         1

The complete graph on six vertices.

         0   24   80  120   120  96   64   35   15   5   1
        24  106  145  105   60   24    6
        50   90   45   15
        35   20
        10
         1

The complete graph on seven vertices.

    0 120  490  945 1225 1260 1120 895 645 420 245 126 56 21 6 1
  120 644 1225 1330 1085  756  469 245 105  35   7
  274 721  700  420  210   84   21
  225 280  105   35
   85  35
   15
    1

The Petersen graph.

    0   36   84   75   35    9    1
   36  168  171   65   10
  120  240  105   15
  180  170   30
  170   70
  114   12
   56
   21
    6
    1

maple has a function called tuttepoly in the networks package.

Last modified January 23, 1996, by S.C. Locke. How to contact me.