The characteristic polynomial of a matroid is a polynomial $p\left(k\right)$ in one variable.

It is an evaluation of the Tutte polynomial $T\left(x\,y\right)$ of that matroid, namely, $p\left(k\right)={\left(\mathrm{-1}\right)}^{\mathrm{rank}\left(M\right)}T\left(1-k\,0\right)$.

Consequently, it respects the operations of deletion and contraction in the same way as the Tutte polynomial.

When applied to a graphic matroid $M\left(G\right)$ associated to a graph $G$, the characteristic polynomial of $M\left(G\right)$ is related to the chromatic polynomial of $G$. Namely, the characteristic polynomial of $M\left(G\right)$ is $k^c$ (where $c$ is the number of connected components of $G$) times the chromatic polynomial of $G$.

$\mathrm{with}\left(\mathrm{Matroids}\right)\:$

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(\left\{\left\{a\,x\right\}\,\left\{a\,y\right\}\,\left\{b\,x\right\}\,\left\{b\,y\right\}\,\left\{c\,x\right\}\,\left\{c\,y\right\}\right\}\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$\mathrm{Chrome}≔\mathrm{ChromaticPolynomial}\left(G\,k\right)$

$\mathrm{Chrome}≔k\left(k-1\right)\left(k^3-5k^2+10k-7\right)$

$M≔\mathrm{Matroid}\left(G\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; graphic\; matroi\ⅆ\; on\; th\ⅇ\; graph:}\\ \mathrm{PLOT}\left(\mathrm{...}\right)\end{array}⟩$

$C≔\mathrm{Matroids}:-\mathrm{CharacteristicPolynomial}\left(M\,k\right)$

$C≔k^4-6k^3+15k^2-17k+7$

$\mathrm{expand}\left(\mathrm{Chrome}-kC\right)$

$0$

James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.