Editing Matroid (section)

===Characteristic polynomial===
The ''[[characteristic polynomial]]'' of ''M'' – sometimes called the ''chromatic polynomial'',<ref name=Wh87127/> although it does not count colorings – is defined to be
:<math>p_M(\lambda) := \sum_{S \subseteq E} (-1)^{|S|}\lambda^{r(E)-r(S)},</math>
or equivalently (as long as the empty set is closed in ''M'') as
:<math>p_M(\lambda) := \sum_{A} \mu(\emptyset,A) \lambda^{r(E)-r(A)} </math>,
where μ denotes the [[Möbius function (combinatorics)|Möbius function]] of the [[geometric lattice]] of the matroid and the sum is taken over all the flats A of the matroid.<ref name=Wh87120>{{harvp|White|1987|p=120}}</ref>

* When ''M'' is the cycle matroid ''M''(''G'') of a graph ''G'', the characteristic polynomial is a slight transformation of the [[chromatic polynomial]], which is given by χ<sub>''G''</sub>&nbsp;(λ) = λ<sup>c</sup>''p''<sub>''M''(''G'')</sub>&nbsp;(λ), where ''c'' is the number of connected components of ''G''.

* When ''M'' is the bond matroid ''M''*(''G'') of a graph ''G'', the characteristic polynomial equals the [[Tutte polynomial#Flow polynomial|flow polynomial]] of ''G''.

* When ''M'' is the matroid ''M''(''A'') of an [[Arrangement of hyperplanes|arrangement]] ''A'' of linear hyperplanes in {{math|ℝ{{sup|n}}}} (or ''F''<sup>''n''</sup> where ''F'' is any field), the characteristic polynomial of the arrangement is given by ''p''<sub>''A''</sub>&nbsp;(λ) = λ<sup>''n''&minus;''r''(''M'')</sup>''p''<sub>''M''</sub>&nbsp;(λ).

====Beta invariant====
The ''beta invariant'' of a matroid, introduced by [[Henry Crapo (mathematician)|Crapo]] (1967), may be expressed in terms of the characteristic polynomial <math> p </math> as an evaluation of the derivative<ref name=Wh87123>{{harvp|White|1987|p=123}}</ref>
:<math> \beta(M) = (-1)^{r(M)-1} p_M'(1) </math>
or directly as<ref name=Wh87124>{{harvp|White|1987|p=124}}</ref>
:<math> \beta(M) = (-1)^{r(M)} \sum_{X \subseteq E} (-1)^{|X|} r(X)</math>.
The beta invariant is non-negative, and is zero if and only if <math> M </math> is disconnected, or empty, or a loop.  Otherwise it depends only on the lattice of flats of <math> M</math>. If <math> M </math> has no loops and coloops then <math> \beta( M ) = \beta( M^* )</math>.<ref name=Wh87124/>