Matroid

In terms of independence, a finite matroid ${\displaystyle M}$ is a pair ${\displaystyle (E,{\mathcal {I}})}$, where ${\displaystyle E}$ is a finite set (called the ground set) and ${\displaystyle {\mathcal {I}}}$ is a family of subsets of ${\displaystyle E}$ (called the independent sets) with the following properties:[4]

(I1) The empty set is independent, i.e., ${\displaystyle \emptyset \in {\mathcal {I}}}$. Alternatively, at least one subset of ${\displaystyle E}$ is independent, i.e., ${\displaystyle {\mathcal {I}}\neq \emptyset }$.

(I2) Every subset of an independent set is independent, i.e., for each ${\displaystyle A'\subseteq A\subseteq E}$, if ${\displaystyle A\in {\mathcal {I}}}$ then ${\displaystyle A'\in {\mathcal {I}}}$. This is sometimes called the hereditary property, or the downward-closed property.

(I3) If ${\displaystyle A}$ and ${\displaystyle B}$ are two independent sets (i.e., each set is independent) and ${\displaystyle A}$ has more elements than ${\displaystyle B}$, then there exists ${\displaystyle x\in A\backslash B}$ such that ${\displaystyle B\cup \{x\}}$ is in ${\displaystyle {\mathcal {I}}}$. This is sometimes called the augmentation property or the independent set exchange property.

The first two properties define a combinatorial structure known as an independence system (or abstract simplicial complex).

A subset of the ground set ${\displaystyle E}$ that is not independent is called dependent. A maximal independent set—that is, an independent set that becomes dependent on adding any element of ${\displaystyle E}$—is called a basis for the matroid. A circuit in a matroid ${\displaystyle M}$ is a minimal dependent subset of ${\displaystyle E}$—that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs.[4]

The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely: a set is independent if and only if it is not dependent, if and only if it is a subset of a basis, and if and only if it does not contain a circuit. The collections of dependent sets, of bases, and of circuits each have simple properties that may be taken as axioms for a matroid. For instance, one may define a matroid ${\displaystyle M}$ to be a pair ${\displaystyle (E,{\mathcal {B}})}$, where ${\displaystyle E}$ is a finite set as before and ${\displaystyle {\mathcal {B}}}$ is a collection of subsets of ${\displaystyle E}$, called "bases", with the following properties:[4]

(B1) ${\displaystyle {\mathcal {B}}}$ is nonempty.

(B2) If ${\displaystyle A}$ and ${\displaystyle B}$ are distinct members of ${\displaystyle {\mathcal {B}}}$ and ${\displaystyle a\in A\setminus B}$, then there exists an element ${\displaystyle b\in B\setminus A}$ such that ${\displaystyle (A\setminus \{a\})\cup \{b\}\in {\mathcal {B}}}$. This property is called the basis exchange property.

It follows from the basis exchange property that no member of ${\displaystyle {\mathcal {B}}}$ can be a proper subset of another.

It is a basic result of matroid theory, directly analogous to a similar theorem of bases in linear algebra, that any two bases of a matroid ${\displaystyle M}$ have the same number of elements. This number is called the rank of ${\displaystyle M}$. If ${\displaystyle M}$ is a matroid on ${\displaystyle E}$, and ${\displaystyle A}$ is a subset of ${\displaystyle E}$, then a matroid on ${\displaystyle A}$ can be defined by considering a subset of ${\displaystyle A}$ to be independent if and only if it is independent in ${\displaystyle M}$. This allows us to talk about submatroids and about the rank of any subset of ${\displaystyle E}$. The rank of a subset ${\displaystyle A}$ is given by the rank function ${\displaystyle r(A)}$ of the matroid, which has the following properties:[4]

• The value of the rank function is always a non-negative integer.
• For any subset ${\displaystyle A\subset E}$, we have ${\displaystyle r(A)\leq |A|}$.
• For any two subsets ${\displaystyle A,B\subset E}$, we have: ${\displaystyle r(A\cup B)+r(A\cap B)\leq r(A)+r(B)}$. That is, the rank is a submodular function.
• For any set ${\displaystyle A}$ and element ${\displaystyle x}$, we have: ${\displaystyle r(A)\leq r(A\cup \{x\})\leq r(A)+1}$. From the first inequality it follows more generally that, if ${\displaystyle A\subseteq B\subseteq E}$, then ${\displaystyle r(A)\leq r(B)\leq r(E)}$. That is, rank is a monotonic function.

These properties can be used as one of the alternative definitions of a finite matroid: if ${\displaystyle (E,r)}$ satisfies these properties, then the independent sets of a matroid over ${\displaystyle E}$ can be defined as those subsets ${\displaystyle A}$ of ${\displaystyle E}$ with ${\displaystyle r(A)=|A|}$. In the language of partially ordered sets, such a matroid structure is equivalent to the geometric lattice whose elements are the subsets ${\displaystyle A\subset M}$, partially ordered by inclusion.

The difference ${\displaystyle |A|-r(A)}$ is called the nullity of the subset ${\displaystyle A}$. It is the minimum number of elements that must be removed from ${\displaystyle A}$ to obtain an independent set. The nullity of ${\displaystyle E}$ in ${\displaystyle M}$ is called the nullity of ${\displaystyle M}$. The difference ${\displaystyle r(E)-r(A)}$ is sometimes called the corank of the subset ${\displaystyle A}$.

Let ${\displaystyle M}$ be a matroid on a finite set ${\displaystyle E}$, with rank function ${\displaystyle r}$ as above. The closure (or span) ${\displaystyle \operatorname {cl} (A)}$ of a subset ${\displaystyle A}$ of ${\displaystyle E}$ is the set

${\displaystyle \operatorname {cl} (A)={\Bigl \{}x\in E\mid r(A)=r{\bigl (}A\cup \{x\}{\bigr )}{\Bigr \}}}$.

This defines a closure operator ${\displaystyle \operatorname {cl}$ :{\mathcal {P}}(E)\to {\mathcal {P}}(E)} where ${\displaystyle {\mathcal {P}}}$ denotes the power set, with the following properties:

• For all subsets ${\displaystyle X}$ of ${\displaystyle E}$, ${\displaystyle X\subseteq \operatorname {cl} (X)}$.
• For all subsets ${\displaystyle X}$ of ${\displaystyle E}$, ${\displaystyle \operatorname {cl} (X)=\operatorname {cl} (\operatorname {cl} (X))}$.
• For all subsets ${\displaystyle X}$ and ${\displaystyle Y}$ of ${\displaystyle E}$ with ${\displaystyle X\subseteq Y}$, ${\displaystyle \operatorname {cl} (X)\subseteq \operatorname {cl} (Y)}$.
• For all elements ${\displaystyle a}$, and ${\displaystyle b}$ of ${\displaystyle E}$ and all subsets ${\displaystyle Y}$ of ${\displaystyle E}$, if ${\displaystyle a\in \operatorname {cl} (Y\cup \{b\})\setminus \operatorname {cl} (Y)}$ then ${\displaystyle b\in \operatorname {cl} (Y\cup \{a\})\setminus \operatorname {cl} (Y)}$.

The first three of these properties are the defining properties of a closure operator. The fourth is sometimes called the Mac Lane–Steinitz exchange property. These properties may be taken as another definition of matroid: every function ${\displaystyle \operatorname {cl}$ :{\mathcal {P}}(E)\to {\mathcal {P}}(E)} that obeys these properties determines a matroid.[4]

A set whose closure equals itself is said to be closed, or a flat or subspace of the matroid.[5] A set is closed if it is maximal for its rank, meaning that the addition of any other element to the set would increase the rank. The closed sets of a matroid are characterized by a covering partition property:

• The whole point set ${\displaystyle E}$ is closed.
• If ${\displaystyle S}$ and ${\displaystyle T}$ are flats, then ${\displaystyle S\cap T}$ is a flat.
• If ${\displaystyle S}$ is a flat, then each element of ${\displaystyle E\setminus S}$ is in precisely one of the flats ${\displaystyle T}$ that cover ${\displaystyle S}$ (meaning that ${\displaystyle T}$ properly contains ${\displaystyle S}$ but there is no flat ${\displaystyle U}$ between ${\displaystyle S}$ and ${\displaystyle T}$).

The class ${\displaystyle {\mathcal {L}}(M)}$ of all flats, partially ordered by set inclusion, forms a matroid lattice. Conversely, every matroid lattice ${\displaystyle L}$ forms a matroid over its set ${\displaystyle E}$ of atoms under the following closure operator: for a set ${\displaystyle S}$ of atoms with join ${\displaystyle \bigvee S}$,

${\displaystyle \operatorname {cl} (S)=\{x\in E\mid x\leq \bigvee S\}}$.

The flats of this matroid correspond one-for-one with the elements of the lattice; the flat corresponding to lattice element ${\displaystyle y}$ is the set

${\displaystyle \{x\in E\mid x\leq y\}}$.

Thus, the lattice of flats of this matroid is naturally isomorphic to ${\displaystyle L}$.

In a matroid of rank ${\displaystyle r}$, a flat of rank ${\displaystyle r-1}$ is called a hyperplane. (Hyperplanes are also called coatoms or copoints.) These are the maximal proper flats; that is, the only superset of a hyperplane that is also a flat is the set ${\displaystyle E}$ of all the elements of the matroid. An equivalent definition is that a coatom is a subset of E that does not span M, but such that adding any other element to it does make a spanning set.[6]

The family ${\displaystyle {\mathcal {H}}}$ of hyperplanes of a matroid has the following properties, which may be taken as yet another axiomatization of matroids:[6]

• There do not exist distinct sets ${\displaystyle X}$ and ${\displaystyle Y}$ in ${\displaystyle {\mathcal {H}}}$ with ${\displaystyle X\subseteq Y}$. That is, the hyperplanes form a Sperner family.
• For every ${\displaystyle x\in E}$ and distinct ${\displaystyle Y,Z\in {\mathcal {H}}}$ with ${\displaystyle x\notin Y\cup Z}$, there exists ${\displaystyle X\in {\mathcal {H}}}$ with ${\displaystyle (Y\cap Z)\cup \{x\}\subseteq X}$.

Minty (1966) defined a graphoid as a triple ${\displaystyle (L,C,D)}$ in which ${\displaystyle C}$ and ${\displaystyle D}$ are classes of nonempty subsets of ${\displaystyle L}$ such that

• no element of ${\displaystyle C}$ (called a "circuit") contains another,
• no element of ${\displaystyle D}$ (called a "cocircuit") contains another,
• no set in ${\displaystyle C}$ and set in ${\displaystyle D}$ intersect in exactly one element, and
• whenever ${\displaystyle L}$ is represented as the disjoint union of subsets ${\displaystyle R,G,B}$ with ${\displaystyle G=\{g\}}$ (a singleton set), then either an ${\displaystyle X\in C}$ exists such that ${\displaystyle g\in X\subseteq R\cup G}$ or a ${\displaystyle Y\in D}$ exists such that ${\displaystyle g\in Y\subseteq B\cup G.}$

He proved that there is a matroid for which ${\displaystyle C}$ is the class of circuits and ${\displaystyle D}$ is the class of cocircuits. Conversely, if ${\displaystyle C}$ and ${\displaystyle D}$ are the circuit and cocircuit classes of a matroid ${\displaystyle M}$ with ground set ${\displaystyle E}$, then ${\displaystyle (E,C,D)}$ is a graphoid. Thus, graphoids give a self-dual cryptomorphic axiomatization of matroids.

Let ${\displaystyle E}$ be a finite set. The set of all subsets of ${\displaystyle E}$ satisfies the definition of a matroid. It is called the free matroid over ${\displaystyle E}$.

Let ${\displaystyle E}$ be a finite set and ${\displaystyle k}$ a natural number. One may define a matroid on ${\displaystyle E}$ by taking every ${\displaystyle k}$-element subset of ${\displaystyle E}$ to be a basis. This is known as the uniform matroid of rank ${\displaystyle k}$. A uniform matroid with rank ${\displaystyle k}$ and with ${\displaystyle n}$ elements is denoted ${\displaystyle U_{k,n}}$. All uniform matroids of rank at least 2 are simple (see § Additional terminology) . The uniform matroid of rank 2 on ${\displaystyle n}$ points is called the ${\displaystyle n}$-point line. A matroid is uniform if and only if it has no circuits of size less than one plus the rank of the matroid. The direct sums of uniform matroids are called partition matroids.

In the uniform matroid ${\displaystyle U_{0,n}}$, every element is a loop (an element that does not belong to any independent set), and in the uniform matroid ${\displaystyle U_{n,n}}$, every element is a coloop (an element that belongs to all bases). The direct sum of matroids of these two types is a partition matroid in which every element is a loop or a coloop; it is called a discrete matroid. An equivalent definition of a discrete matroid is a matroid in which every proper, non-empty subset of the ground set ${\displaystyle E}$ is a separator.

The Fano matroid, derived from the Fano plane. It is GF(2)-linear but not real-linear.

The Vámos matroid, not linear over any field

Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces. There are two ways to present the matroids defined in this way:

• If ${\displaystyle E}$ is any finite subset of a vector space ${\displaystyle V}$, then we can define a matroid ${\displaystyle M}$ on ${\displaystyle E}$ by taking the independent sets of ${\displaystyle M}$ to be the linearly independent subsets of ${\displaystyle E}$. The validity of the independent-set axioms for this matroid follows from the Steinitz exchange lemma. If ${\displaystyle M}$ is a matroid that can be defined in this way, we say the set ${\displaystyle E}$ represents ${\displaystyle M}$. Matroids of this kind are called vector matroids. An important example of a matroid defined in this way is the Fano matroid, a rank-three matroid derived from the Fano plane, a finite geometry with seven points (the seven elements of the matroid) and seven lines (the proper nontrivial flats of the matroid). It is a linear matroid whose elements may be described as the seven nonzero points in a three-dimensional vector space over the finite field GF(2). However, it is not possible to provide a similar representation for the Fano matroid using the real numbers in place of GF(2).
• A matrix ${\displaystyle A}$ with entries in a field gives rise to a matroid ${\displaystyle M}$ on its set of columns. The dependent sets of columns in the matroid are those that are linearly dependent as vectors. This matroid is called the column matroid of ${\displaystyle A}$, and ${\displaystyle A}$ is said to represent ${\displaystyle M}$. For instance, the Fano matroid can be represented in this way as a 3 × 7 (0,1)-matrix. Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation. (There is one technical difference: a column matroid can have distinct elements that are the same vector, but a vector matroid as defined above cannot. Usually this difference is insignificant and can be ignored, but by letting ${\displaystyle E}$ be a multiset of vectors one brings the two definitions into complete agreement.)

A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable or linear. If ${\displaystyle M}$ is equivalent to a vector matroid over a field ${\displaystyle F}$, then we say ${\displaystyle M}$ is representable over ${\displaystyle F}$; in particular, ${\displaystyle M}$ is real-representable if it is representable over the real numbers. For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroid theory is to characterize the matroids that may be represented over a given field ${\displaystyle F}$; Rota's conjecture describes a possible characterization for every finite field. The main results so far are characterizations of binary matroids (those representable over GF(2)) due to Tutte (1950s), of ternary matroids (representable over the 3-element field) due to Reid and Bixby, and separately to Seymour (1970s), and of quaternary matroids (representable over the 4-element field) due to Geelen, Gerards, and Kapoor (2000). This is very much an open area.

A regular matroid is a matroid that is representable over all possible fields. The Vámos matroid is the simplest example of a matroid that is not representable over any field.

A second original source for the theory of matroids is graph theory.

Every finite graph (or multigraph) ${\displaystyle G}$ gives rise to a matroid ${\displaystyle M(G)}$ as follows: take as ${\displaystyle E}$ the set of all edges in ${\displaystyle G}$ and consider a set of edges independent if and only if it is a forest; that is, if it does not contain a simple cycle. Then ${\displaystyle M(G)}$ is called a cycle matroid. Matroids derived in this way are graphic matroids. Not every matroid is graphic, but all matroids on three elements are graphic.[7] Every graphic matroid is regular.

Other matroids on graphs were discovered subsequently:

• The bicircular matroid of a graph is defined by calling a set of edges independent if every connected subset contains at most one cycle.
• In any directed or undirected graph ${\displaystyle G}$ let ${\displaystyle E}$ and ${\displaystyle F}$ be two distinguished sets of vertices. In the set ${\displaystyle E}$, define a subset ${\displaystyle U}$ to be independent if there are |${\displaystyle U}$| vertex-disjoint paths from ${\displaystyle F}$ onto ${\displaystyle U}$. This defines a matroid on ${\displaystyle E}$ called a gammoid:[8] a strict gammoid is one for which the set ${\displaystyle E}$ is the whole vertex set of ${\displaystyle G}$.[9]
• In a bipartite graph ${\displaystyle G=(U,V,E)}$, one may form a matroid in which the elements are vertices on one side ${\displaystyle U}$ of the bipartition, and the independent subsets are sets of endpoints of matchings of the graph. This is called a transversal matroid,[10][11] and it is a special case of a gammoid.[8] The transversal matroids are the dual matroids to the strict gammoids.[9]
• Graphic matroids have been generalized to matroids from signed graphs, gain graphs, and biased graphs. A graph ${\displaystyle G}$ with a distinguished linear class ${\displaystyle B}$ of cycles, known as a "biased graph" ${\displaystyle (G,B)}$, has two matroids, known as the frame matroid and the lift matroid of the biased graph. If every cycle belongs to the distinguished class, these matroids coincide with the cycle matroid of ${\displaystyle G}$. If no cycle is distinguished, the frame matroid is the bicircular matroid of ${\displaystyle G}$. A signed graph, whose edges are labeled by signs, and a gain graph, which is a graph whose edges are labeled orientably from a group, each give rise to a biased graph and therefore have frame and lift matroids.
• The Laman graphs form the bases of the two-dimensional rigidity matroid, a matroid defined in the theory of structural rigidity.
• Let ${\displaystyle G}$ be a connected graph and ${\displaystyle E}$ be its edge set. Let ${\displaystyle I}$ be the collection of subsets ${\displaystyle F}$ of ${\displaystyle E}$ such that ${\displaystyle G-F}$ is still connected. Then ${\displaystyle M^{*}(G)}$, whose element set is ${\displaystyle E}$ and with ${\displaystyle I}$ as its class of independent sets, is a matroid called the bond matroid of ${\displaystyle G}$. The rank function ${\displaystyle r(F)}$ is the cyclomatic number of the subgraph induced on the edge subset ${\displaystyle F}$, which equals the number of edges outside a maximal forest of that subgraph, and also the number of independent cycles in it.

A third original source of matroid theory is field theory.

An extension of a field gives rise to a matroid. Suppose ${\displaystyle F}$ and ${\displaystyle K}$ are fields with ${\displaystyle K}$ containing ${\displaystyle F}$. Let ${\displaystyle E}$ be any finite subset of ${\displaystyle K}$. Define a subset ${\displaystyle S}$ of ${\displaystyle E}$ to be algebraically independent if the extension field ${\displaystyle F(S)}$ has transcendence degree equal to ${\displaystyle |S|}$.[12]

A matroid that is equivalent to a matroid of this kind is called an algebraic matroid.[13] The problem of characterizing algebraic matroids is extremely difficult; little is known about it. The Vámos matroid provides an example of a matroid that is not algebraic.

If M is a finite matroid, we can define the orthogonal or dual matroid M* by taking the same underlying set and calling a set a basis in M* if and only if its complement is a basis in M. It is not difficult to verify that M* is a matroid and that the dual of M* is M.[14]

The dual can be described equally well in terms of other ways to define a matroid. For instance:

• A set is independent in M* if and only if its complement spans M.
• A set is a circuit of M* if and only if its complement is a coatom in M.
• The rank function of the dual is ${\displaystyle r^{*}(S)=|S|-r(M)+r\left(E\setminus S\right)}$.

According to a matroid version of Kuratowski's theorem, the dual of a graphic matroid M is a graphic matroid if and only if M is the matroid of a planar graph. In this case, the dual of M is the matroid of the dual graph of G.[15] The dual of a vector matroid representable over a particular field F is also representable over F. The dual of a transversal matroid is a strict gammoid and vice versa.

Example

The cycle matroid of a graph is the dual matroid of its bond matroid.

If M is a matroid with element set E, and S is a subset of E, the restriction of M to S, written M |S, is the matroid on the set S whose independent sets are the independent sets of M that are contained in S. Its circuits are the circuits of M that are contained in S and its rank function is that of M restricted to subsets of S. In linear algebra, this corresponds to restricting to the subspace generated by the vectors in S. Equivalently if T = M−S this may be termed the deletion of T, written M\T or M−T. The submatroids of M are precisely the results of a sequence of deletions: the order is irrelevant.[16][17]

The dual operation of restriction is contraction.[18] If T is a subset of E, the contraction of M by T, written M/T, is the matroid on the underlying set E − T whose rank function is ${\displaystyle r'(A)=r(A\cup T)-r(T).}$[19] In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in T, together with the images of the vectors in E - T.

A matroid N that is obtained from M by a sequence of restriction and contraction operations is called a minor of M.[17][20] We say M contains N as a minor. Many important families of matroids may be characterized by the minor-minimal matroids that do not belong to the family; these are called forbidden or excluded minors.[21]

Let M be a matroid with an underlying set of elements E, and let N be another matroid on an underlying set F. The direct sum of matroids M and N is the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an independent set of N.

The union of M and N is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whose independent sets are those subsets that are the union of an independent set in M and one in N. Usually the term "union" is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the direct sum.

A weighted matroid is a matroid together with a function from its elements to the nonnegative real numbers. The weight of a subset of elements is defined to be the sum of the weights of the elements in the subset. The greedy algorithm can be used to find a maximum-weight basis of the matroid, by starting from the empty set and repeatedly adding one element at a time, at each step choosing a maximum-weight element among the elements whose addition would preserve the independence of the augmented set.[29] This algorithm does not need to know anything about the details of the matroid's definition, as long as it has access to the matroid through an independence oracle, a subroutine for testing whether a set is independent.

This optimization algorithm may be used to characterize matroids: if a family F of sets, closed under taking subsets, has the property that, no matter how the sets are weighted, the greedy algorithm finds a maximum-weight set in the family, then F must be the family of independent sets of a matroid.[30]

The notion of matroid has been generalized to allow for other types of sets on which a greedy algorithm gives optimal solutions; see greedoid and matroid embedding for more information.

The matroid partitioning problem is to partition the elements of a matroid into as few independent sets as possible, and the matroid packing problem is to find as many disjoint spanning sets as possible. Both can be solved in polynomial time, and can be generalized to the problem of computing the rank or finding an independent set in a matroid sum.

The intersection of two or more matroids is the family of sets that are simultaneously independent in each of the matroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids can be found in polynomial time, and provides a solution to many other important combinatorial optimization problems. For instance, maximum matching in bipartite graphs can be expressed as a problem of intersecting two partition matroids. However, finding the largest set in an intersection of three or more matroids is NP-complete.

Two standalone systems for calculations with matroids are Kingan's Oid and Hlineny's Macek. Both of them are open sourced packages. "Oid" is an interactive, extensible software system for experimenting with matroids. "Macek" is a specialized software system with tools and routines for reasonably efficient combinatorial computations with representable matroids.

Both open source mathematics software systems SAGE and Macaulay2 contain matroid packages.

The characteristic polynomial of M (which is sometimes called the chromatic polynomial,[31] although it does not count colorings), is defined to be

${\displaystyle p_{M}(\lambda ):=\sum _{S\subseteq E}(-1)^{|S|}\lambda ^{r(M)-r(S)},}$

or equivalently (as long as the empty set is closed in M) as

${\displaystyle p_{M}(\lambda ):=\sum _{A}\mu (\emptyset ,A)\lambda ^{r(M)-r(A)}\ ,}$

where μ denotes the Möbius function of the geometric lattice of the matroid and the sum is taken over all the flats A of the matroid.[32]

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_G (λ) = λ^cp_M(G) (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 flow polynomial of G.

When M is the matroid M(A) of an arrangement A of linear hyperplanes in Rⁿ (or Fⁿ where F is any field), the characteristic polynomial of the arrangement is given by p_A (λ) = λ^n−r(M)p_M(A) (λ).

The beta invariant of a matroid, introduced by Crapo (1967), may be expressed in terms of the characteristic polynomial p as an evaluation of the derivative[33]

${\displaystyle \beta (M)=(-1)^{r(M)-1}p_{M}'(1)\ }$

or directly as[34]

${\displaystyle \beta (M)=(-1)^{r(M)}\sum _{X\subseteq E}(-1)^{|X|}r(X)\ .}$

The beta invariant is non-negative, and is zero if and only if M is disconnected, or empty, or a loop. Otherwise it depends only on the lattice of flats of M. If M has no loops and coloops then β(M) = β(M^∗).[34]

The Tutte polynomial of a matroid, T_M (x,y), generalizes the characteristic polynomial to two variables. This gives it more combinatorial interpretations, and also gives it the duality property

${\displaystyle T_{M^{*}}(x,y)=T_{M}(y,x),}$

which implies a number of dualities between properties of M and properties of M *. One definition of the Tutte polynomial is

${\displaystyle T_{M}(x,y)=\sum _{S\subseteq E}(x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.}$

This expresses the Tutte polynomial as an evaluation of the corank-nullity or rank generating polynomial,[35]

${\displaystyle R_{M}(u,v)=\sum _{S\subseteq E}u^{r(M)-r(S)}v^{|S|-r(S)}.}$

From this definition it is easy to see that the characteristic polynomial is, up to a simple factor, an evaluation of T_M, specifically,

${\displaystyle p_{M}(\lambda )=(-1)^{r(M)}T_{M}(1-\lambda ,0).}$

Another definition is in terms of internal and external activities and a sum over bases, reflecting the fact that T(1,1) is the number of bases.[36] This, which sums over fewer subsets but has more complicated terms, was Tutte's original definition.

There is a further definition in terms of recursion by deletion and contraction.[37] The deletion-contraction identity is

${\displaystyle F(M)=F(M-e)+F(M/e)}$ when ${\displaystyle e}$ is neither a loop nor a coloop.

An invariant of matroids (i.e., a function that takes the same value on isomorphic matroids) satisfying this recursion and the multiplicative condition

${\displaystyle F(M\oplus M')=F(M)F(M')}$

is said to be a Tutte-Grothendieck invariant.[35] The Tutte polynomial is the most general such invariant; that is, the Tutte polynomial is a Tutte-Grothendieck invariant and every such invariant is an evaluation of the Tutte polynomial.[31]

The Tutte polynomial T_G  of a graph is the Tutte polynomial T_M(G) of its cycle matroid.