characteristicPolynomial(Matroid) -- computes characteristic polynomial of a matroid

Synopsis

Function: characteristicPolynomial
Usage:

characteristicPolynomial M
Inputs:
- M, a matroid,
Optional inputs:
- VariableName => ..., default value q,
Outputs:
- a ring element, the characteristic polynomial of M

Description

The characteristic polynomial is a particular specialization of the Tutte polynomial. If M is a matroid of rank r with Tutte polynomial T(x, y), then the characteristic polynomial of M is given by (-1)^r * T(1 - x, 0).

This function computes the characteristic polynomial as an evaluation of the Tutte polynomial. If the Tutte polynomial of the matroid has already been computed, then this function should return the characteristic polynomial instantaneously.

i1 : M = matroid completeGraph 4

o1 = a "matroid" of rank 3 on 6 elements

o1 : Matroid

i2 : T = tuttePolynomial M

      3    3     2            2
o2 = x  + y  + 3x  + 4x*y + 3y  + 2x + 2y

o2 : ZZ[x..y]

i3 : factor characteristicPolynomial M

o3 = (x - 3)(x - 2)(x - 1)

o3 : Expression of class Product

Caveat

If M = M(G) is a graphic matroid, then the characteristic polynomial of M and the chromatic polynomial of G differ by a factor of x^k, where k is the number of connected components of the graph G.

Ways to use this method: